DMACC Pre-Calc Final
The Transverse Axis =
2a
Minor axis =
2b
State the Vertex and Leading Coefficent: f(x) = 2(x-1)^2 - 3
(1, -3) LC: 2
Find all polar coordinates of polar coordinate (1.5, -20 degrees) for -2pi <= x <=2pi
(1.5 , -pi/9) (-1.5, -10pi/9) (-1.5, 8pi/9) (1.5, 17pi/9)
Foci in an ellipse
(c, 0) and (-c,0) relative to the center
i^2 =
-1
Plotting points with given polar coordinate (3, 4pi/3)
1. Draw picture and find 4pi/3 3rd quadrant 60 angle 2. Then go magnitude from 0 length of 3
Identify max r value r = 2 + 3cosx
5
Rules for Synthetic and Long division
1. Must be written in standard form 2. All terms must be present
Rowing speed and Current speed problems
1. Remember that d = r*t 2. Find the rate ( r = d/t --> Speed) -Let R be rowing speed -Let C be current speed 3. With the current/stream: R + C = r --> R + C = d/t 4. Against the stream: R - C = r --> R - C = d/t 5. Use both equations to get the same variable by itself and substitute to solve for the other 6. Repeat step 5 for the other variable
To solve for a circle ...
1. Set the equation = y x^2 + y^2 = # y^2 = # - x^2 y = + - sqrt# - x^2) 2. Simplify and you will have two equations to plug into the calculator
Given 7(cos 135i + sin135j) State the magnitude and direction angle
7 @ 135 degrees
Simplify (12 : 7)
792
A geometric sequence {an} begins 2, 6, ... What is a6/a2?
81
Evaluate the expression 7P4
840
Covert polar equation to rectangular equation and identify the graph r= -3 sin(x)
1. When cos or sin is used multiply by r r *r = -3 r sin(x) r^2 = -3rsin(x) 2. r^2 turns into x^2 + y^2 and rsin(x) = y x^2 + y^2 = -3y 3. Add 3y x^2 + y^2 +3y ______ = 0 4. Complete the square x^2 + (y + 3/2)^2 = 9/4 A circle at center ( 0, -3/2) with radius of (3/2)
Covert polar equation to rectangular equation and identify the graph r= 3 sec(x)
1. When sec or csc are used multiply by cos or sin respectivly r * cos(x) = 3sec(x) * cos(x) x = 3 Vertical line at x = 3
Keyword for combinations
"Get"
Find the sum of the geometric sequence: 3, 6, 12, ... , 3072
11Ek=1(3*2^k-1) = 6141
Find the center, foci, and verticies of the ellipse (x-1)^2/36 + (y+5)^2/16 = 1
C:(1,-5) F: (1 + 2sqrt(5),-5), (1-2sqrt(5), -5) V:(7,-5), (-5,-5)
In a Polynomial the leading coeficcents ...
Can not be 0
Identify the graph r = 4+4cosx
Cartiod
Strength of a scatterplot is based on
If the points form a line closer to a line = strong more spread out = weak
Two vectors are parallel ...
If they have the same slope.
The coefficent of the term with the highest degree in a polynomial is called what?
Leading Coefficent
Describe how to transform the graph x^2 to the given function. f(x) = 1/2(x+2)^2 - 3
Left 2 Down 3 Vertical Shrink of magnitude 1/2
Identify the graph r^2 = sin2x; 0<= x <= 2pi
Lemniscate
Determine if the following is a polynomial function. If it is then state the degree and leading coefficent; if it is not then explain why not. f(x) = 3x^-5 + 17
Not a Polynomial It has a negative exponent; Polynomial Functions only have Positive Integer exponents
Geometric series formula
Sn = a1(1 - r^n)/1-r
The first row of seats in section J of the stadium has 10 seats. In all there are 7 rows of seats in section J with each row containing 4 more seats than the preceding row. How many seats are in section J
There are 154 total seats in section J. Reference #7 sequences 9.6
An identification code is to consist of 3 letters followed by two digits. How many different codes are possible if repetition is permitted.
There are 1757600 different indentification codes following the pattern
Find the verticies and foci of the ellipse 8x^2 + 9y^2 = 72
V:(3,0), (-3,0) F: (-1,0),(1,0)
Find the horizontal and vertical asymptotes of f(x). Use limits to describe the cooresponding behavior. f(x) = 5x^2 + 2 / x^2 + 4
VA: None HA: y = 5 lim x --> -inf f(x) = 5 lim x --> inf f(x) = 5
Vertical asymptote format
VA: x = #
Prove that the graph of the equation is a parabola, and find its vertex, focus, and directrix y^2-8y-8x+24
Vertex(1,4) Focus:(3, 4) Directrix: x = -1
A certain parabola has the equation: (y+4)^2 = -8(x-2) Find the vertex, focus, directrix, and Focal width
Vertex(2,-4) Focus(0, -4) Directrix: x = 4 Focal Width: 8
Solve log x^2 = 2
When dealing with solving, don't use exponential logarithm properties 10^2 = x^2 100 = x^2 x = +- 10
For trig angles only When finding the angle of direction for a vector, you need find the angle inside the triangle and then use these formulas to determine the angle of direction. Use a as the angle inside the triangle and theta as the direction angle of the magnitude
When in Quad 1: theta = a When in Quad 2: theta = 180 - a When in Quad 3: theta = 180 + a When in Quad 4: theta = 360 - a
When asked a ? for # of combinations or permutations
You must write a sentence
Permutations are when ...
You select all the possible combinations and then rearrange them
Combinations are when ...
You select all the possible combinations without rearranging them
Find all real zeros of the function, finding exact values whenever possible, identify each zero as rational or irrational: f(x) = 7x^3 - 2x^2-35x + 10
Zeros: 2/7 rational +/- sqrt(5) irrational
For the following polynomial, one zero is given. Find the remaining zeros: f(x) = x^4 + 7x^2 - 144 Zero: 4i
Zeros: 4i, -4i, 3, -3
Rose Curve
a * cosbx or a * sinbx
An extraneous solution is
a zero that makes the LCD = 0
Factored form format
a(x-k)(x-k)(x-k)
Partial Fraction Decomposition Format
a+b/c*d = ?/c + ?/d
Explicit Formula for Geometric Sequence
an = a1 * r^(n-1)
Explicit Fomrula for an arithmetic sequence
an = a1 + d(n-1)
A parabola wraps ________ the focus and ________ from the directrix
around; away
Eccentricity =
c/a
For a hyperbola to find the last variable given the other two what is the equation to use?
c^2 = a^2 + b^2
limacon: a = b
cartiod
Distance for 3 dimensional points (x1, y1, z1) , (x2, y2, z2)
d = sqrt( (x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2 )
Find the distance between the two points (1, -3, 0) and (0,0,-1)
d = sqrt(11)
Where (x + y)^# x does what when factored out
decreases from left to right
Given an even integer, the function will be an This will cause it to ...
even function; reflect over the y axis
For ellipses a must be
greater than b
Where (x + y)^# y does what when factored out
increases from left to right
Horizontal aysmptote end behavior rules and format
lim x --> inf f(x) = # lim x --> -inf f(x) = # Rules plug in a large value for x and determine if it is a horizontal asymptote by comparing it with the horizontal asymptote test
Identify the graph r = 2 + 5 cos x
limacion with inner loop
a +- bcosx or a+-sinbx
limacon curve
limacon: 0 < a/b < 1
limacon with inner loop
logarithm arguments cannot be
negative
as numbers approach a vertical asymptote from either side and go down the end behavior is ...
negative infinity
In a three dimensional graph dashed lines are
negative values
Given an odd integer, the function will be an This will cause it to...
odd function; Have origin symmetry
When solving a system of equations with 2 variables the answer will be a
ordered pair
When solving a system of equations with 3 variables the answer will be a
ordered triple
Distance from vertex to focus
p
Use the binomial theroem to expand the expression (p + q)^4
p^4 + 4p^3q + 6p^2q^2 + 4pq^3 + q^4
Substitution using triangular form
plug the single variable into the second equation and then both into the first
Series have
plus signs +
Fraction form: Ex: 2x^4-x^3 - 2 ------------- 2x^2 + x + 1
problem = quotient + remainder /divisor Ex: 2x^4-x^3 - 2 ------------- 2x^2 + x + 1 2x^4-x^3 - 2 ------------- = x^2 - x + ( x-2 / 2x^2 + x + 1) 2x^2 + x + 1
spiral of archimedes
r = theta
Calculate the expression using r = <3, 0, -1> and v = <-7, 5, -6> r + v
r+v = <-4, 5, -7>
Lemniscate
r^2 = a^2 * sin2x or r^2 = a^2 * cos2x
r = theta
spiral of archimedes
You find the unit vector of a normal vector by ...
take the vector in component form * 1/magnitude of the vector vector u = 1/|v| * vector v
Determine whether the sequence converges or diverges. If it converges give the limit 1, 4, 9, 16, ...., n^2, ...
the nth term is n^2 so lim n^2 n --> inf. = inf
Multiplying two of the same radical =
the number underneath the radical
In order to add / substract matrices
the order must be the same If it is add/subtract cooresponding positions
Square matrices have
the same number of rows as columns
p and and the directrix equation affect
the same variable
b's exponent is always
what the denominator is in the combination
In a three dimensional graph the x,y,z coordinates go
x --> Diaganolly y --> horizontally z --> Vertically
Write the parametric equations for the line through C and the midpoint of AB A(0, -5, 5) B(10, -11, 9) C(-3, -2, -5)
x = -3 + 8t y = -2 - 6t x = -5 + 12t
The axisof a quadratic function is the
x coordinate of the vertex
when given an exponent of 1/x it is the same as
x root of base
When it is a vertical parabola, p affects the _ value
y
Key to logartithms
y = a *b^x x = log<b> y
Recursive setup for a geometric sequence
{an = #; an+1 = an * r; n >= 1
Keyword for permutations
"Rearrange"
lim x --> #
# is a vertical asymptote
Determine whether the sequence converges or diverges. If it converges give the limit an = (0.7)^n
(#)^n if # > 1 then diverges if # < 1 then converges lim n --> inf = 0
State the Vertex and leading coefficent: f(x) = 2(x+1)^2 - 3
(-1, -3) LC: 2
Find the rectangular coordinate of (-3, -29pi/7)
(-2.70, 1.30)
Base points of an exponential function
(0,1) and (1,b)
State the Vertex and Leading Coefficent: f(x) = 4 - 3(x-1)^2
(1, 4) LC: -3
Base points for logarithmic functions
(1,0) and (b,1)
Find the partial fraction decomposition for the rational expression (20 / x^2-4x)
(20 / x^2-4x) = ( -5 / x) + (5 / x-4)
Find the partial fraction decomposition for the rational expression (7x + 57 / x^2+15x+54 )
(7x + 57 / x^2+15x+54 ) = (5 / x+6) + (2 / x+9)
What is the coefficent of w^5y^3 in the expansion of (w+y)^8
(8 : 3) (w)^5 (y)^3 56
Given a power function formula write a description:
(dependent variable) is directly/inversly proportional to (independent variable), with a constant of variation of (constant)
Find an equation in standard form for the hyperbola that satisfies the given conditions: Foci (-7, 4) and (3, 4) Transverse axis endpoints (-4, 4) and (0, 4)
(x+2)^2/4 - (y-4)^2/21 = 1
Write an equation for the sphere with center (-7, 8, 9) and radius 9
(x+7)^2 + (y-8)^2 + (z-9)^2 = 81
Find an equation in standard form for the hyperbola that satisfies the given conditions: Transverse Axis endpoints (-1, 5) and (5,5) Slope of one asymptote = 4/3
(x-2)^2/9 - (y-5)^2/16 = 1
The equation for a sphere is
(x-h)^2 + (y-k)^2 + (z-L)^2 = r^2
equation for a horizontal hyperbola at any given point
(x-h)^2/a^2 - (y-k)^2/b^2 = 1
The signs of the coefficent are determined by the sign inside ...
+ = all addittion - = alternating + and -
Express the rational number as a fraction of integers: -14.694694694
-14 + .694694694 .694 --> 1/1000 inf E 694(1/1000)^k -14 + [.694 + .000694 + .000000694] .694 / 1 - 1/1000 --> .694 / .999 --> 694/999 + 14 ---> 14680/999 --> -14680/999
Find the coefficent of the given term in the binomial expression: x^2 term, (x-4)^7
-21504
What elementary row operations applied to the first matrix will yeild the second matrix [-3 3 -2 1] [1 -2 3 0] [4 2 3 1] --> [-3 3 -2 1] [1 -2 3 0] [0 10 -9 1]
-4(R2) + R3
Compute the exact value of the function for the given x-value f(x) = -5*2^x for x = 1/3
-5*cbrt(2)
h =
-b/2a
Line segments always have a restriction of...
0<= t <= 1
0! =
1
Exponential graphs always have
1 HA 2 points
Logarithmic graphs always have
1 VA 2 points
State how many complex and real zeros the function has: f(x) = x^2 -10x +41
1. # of complex zeros = degree = 2 2. # of real zeros = plug into graphing calc = 0 3. Sketch graph with viewing window
Find all polar coordinates of polar coordinate (2, pi/6) for -2pi <= x <=2pi
1. (2, pi/6) 2. Draw a picture of scenario 1rst quadrant and postive number 3. (-2, 7pi/6) 4.(2, - 11pi/6) 5.(-2, -5pi/6)
Determine whether B is an inverse of A A = [1 -2] [-1 3] B = [3 2] [1 1]
1. Compare the order of A * B 2x2 2x2(Same rows and columns) 2. Multiply AB = [1(3)+-2(1) 1(2)+-2(1)] [-1(3)+3(1) -1(2)+3(1)] AB = [1 0] [0 1] Yes it is an inverse
To prove an ellipse equation:
1. Complete the square by seperating x and y squared terms
Solve by the elimination method {4x - 3y = 2 {-8x + 6y = -4
1. Eliminate 1 variable 2(4x - 3y) = 2(2) 8x - 6y = 4 -8x + 6y = -4 0 = 0 --> 0 = 0 means infinite solutions 2. Write as a solution set {(x,y) | 4x -3y = 2}
Prove that vector RS and vector PQ are equivilant by showing that they represent the same vector R=(-4,7) S = (-1,5) P = (0,0) Q=(3, -2)
1. Find the component form of RS by finding the distance in the x and distance in the y RS = <-4,7> --> <-1,5> RS = <3, -2> 2. Find the component form of PQ by finding the distance in the x and distance in the y PQ = <0,0> --> <3,-2> PQ = <3,-2> 3. Compare RS = PQ <3,-2> = <3,-2>
Find a parametrization for the curve The line segment with endpoints (5,2) and (-2,-4)
1. Find the slope vector <-7, -6> 2. Plug into formula <x,y> = <-7,-6> * t + (5,2) x = -7t + 5 y = -6t + 2 0<= t <=1
Determine the real values of x that can cause the function to be a) Zero, b) Undefined, c) Positive, d) Negative f(x) = (x-1) / (5x+8)(x-6)
1. Find the zeros (What makes the numerator 0) x-1 = 0 x = 1 2. Undefined (Negative roots, or 0 denominator) 5x+8 = 0 x = -8/5 x-6=0 x=6 3. Place values on sign chart and test regions 4. Determine what makes the function positive f(x) > 0: (-8/5, 1)U(6, inf) 5. Determine what makes the function negative f(x) < 0: (-inf, -8/5)U(1,6)
Solve by the substitution method {y = x^2 {y - 100 = 0
1. Get a variable by itself y - 100 = 0 y = 100 2. Substitute and solve for other variable 100 = x^2 x = +- 10 3. Plug answer back into orginal equation y = (10)^2 --> y = 100 y = (-10)^2 --> y = 100 4. Check answer with other equation 100 - 100 = 0 100 - 100 = 0 5. Write as ordered pair {(-10, 100), (10, 100)}
Determine the number of solutions of the system {4x + 7y = 43 {5x -4y = -10
1. Get equations to slope form 4x + 7y = 43 7y = -4x + 43 y = -4/7x + 43/7 5x - 4y = -10 -4y = -5x -10 y = 5/4x + 10/4 y = 5/4x + 5/2 2. Compare slopes and y-intercepts y = -4/7x + 43/7 y = 5/4x + 5/2 3. Not Same Slopes = Only 1 Intersection
Substitution Rules
1. Get one equation to be a variable by itself 2. Substitute that variable into the other equation 3. Solve for the other variable 4. Plug in other variable for original variable 5. Double check by placing values into orginal equation
To prove a parabola equation:
1. Get the squared variable on 1 side with the opposite variable and constants on other side 1.5 Factor if needed 2. Complete the square -Take second coefficent /2 ^2 and add to each side -Multiply by factor if needed 3. Write as a squared equation with the variable minus the result of second coefficent/2 4. Must be in equation form
Objective Function problems procedure
1. Graph inequalities 2. Identify corners 3. Loacate maximum and minimums if needed
Steps to partial fraction decomposition
1. Idenify common denominator 2. Multiply each term by the denominator and plug in A and B for the missing terms 3. Set x = to a number to cancel out A, and solve for B and then replace it in the equation 4. Repeat 3 but cancel B
Describe how to transform the graph f(x) = x^n into the graph of the given polynomial function: g(x) = -1/2(x+4)^3 + 1; Sketch the transformed graph; Find the y intercept
1. Identify transformations and place them from inside out Left 4, Vertical Shrink of magnitude 1/2, Reflection over the x axis, up 1 2. Find y intercept by plugging 0 in for x -1/2(0+4)^3+1 -1/2(4)^3 +1 -1/2 (64) + 1 -32 + 1 y = -31 yint = (0, -31) 3. Refer to Assignment Polynomials of Higher Degrees for Sketch
Describe how to transform the graph f(x) = x^n into the graph of the given polynomial function: g(x) = 2(x+5)^3; Sketch the transformed graph
1. Identify transformations and place them from inside out Left 5, Vertical Stretch of magnitude 2 2. Refer to Assignment Polynomials of Higher Degrees for Sketch
Use the properties of logarithms to write the expression as a sum or difference of logarithms or multiples of logarithms ln(25d)
1. Identify which propert log(R*S) = log R + log S 2. Split ln 25 + ln d 3. Reduce ln 5^2 + ln d 2 ln 5 + ln d
Describe how to transform the graph of f(x) into the graph of g(x) Sketch the graph by hand and support answer with a grapher f(X) = 3^x g(x) = 3^x-3
1. List out transformations Right 3 units 2. Apply transformations to the base points of an exponential function (0,1) and (1,b) (0,1) --> (3,1) (1,b) --> (1,3) --> (4,3) 3. Graph
Use Gaussian Elimination to solve the linear system by finding an equivilant system in triangular form {5x + 12y + 6z = 14 {2x + 5y + 4z = -9 {x + 2y -2z = 6
1. Move the bottom equation to the top to make life easier {x + 2y -2z = 6 {5x + 12y + 6z = 14 {2x + 5y + 4z = -9 2. Eliminate all x's from other equations -2(x+2y-2z=6) --> -2x -4y + 4z = -12 Add equations together to get a new one (-2x - 4y + 4z = -12) + (2x + 5y + 4z = -9) = y + 8z = -21 ----------------------------------------------- -5(x+2y-2z=6) --> -5x -10y + 10z = -30 (-5x-10y+10z=-30) + (5x+12y+6z=14) = 2y + 16z = -16 3. Rewrite system with new equations {x + 2y -2z = 6 {y + 8z = -21 {2y + 16z = -16 4. Repeat process to eliminate y from last equation -2(y+8z=-21) --> -2y -16z = 42 (-2y -16z = 42) + (2y + 16z = -16) = 0 = 26 No Solution
State whether the function is exponential growth or decay and describe using limits f(x) = 1/3^-x
1. Must covert into a*b^x 1/3^-x 3^x 2. Compare to rules Growth: a>0 b>1 Decay: a>0 0<b<1 Exponential Growth lim x--> -inf f(x) = inf lim x--> inf f(x) = 0
The point vector is
1. The only point given OR 2. The head of the directional vector
Determine a formula for the exponential function with points (0, 2.5) and (1,4.7)
1. Write formula: P(t) = P(1 + r)^t 2. Plug in (0, 2.5) 2.5 = P(1+r)^0 P = 2.5 3. Plug in (1, 4.7) 4.7 = 2.5(1+r)^1 r = .88 4. Write as function P(t) = 2.5(1.88)^t
Write a polynomial function of minimum degree in standard form with real coefficents whose zeros and their multiplicities include those listed: 1 (multiplicity of 2) -5 (multiplicity of 3)
1. Write out all factors (x-1)(x-1)(x+5)(x+5)(x+5) 2. Multiply and Simplify x^5 + 13x^4 + 46x^3 -10x^2 -175x +125
Evaluate the logarithimic expression: log<6> 1296
1. write as exponential 1296 = 6^x 6^4 = 1296 log<6> 1296 = 4
Determine wheter the vectors u and v are parallel, orthogonal, or neither u = <15, -12> v = <-4, 5>
1.Determine Slope slope u = (-12/15) --> (-4/5) slope v = (5/-4) 2. Determine if the vectors are orthogonal by finding the dot product U*v = -120 3. The vectors are neither parallel or orthoganal
When given a parametric equation where x = ...t and y = ...t , and asked for slope intercept form then...
1.Find t and subsitute in for either variable to get the equation to equal only x and ys 2. Write in y = mx +b format
Find the component form of vector v |v| = 18 and Given angle is 25. Reference image on pg 464 #29
1.Find the x component using cos: |v| * cos(theta) x component = 18 * cos(25) x component = 16.313 2. Find the y component using sin: |v| * sin(theta) y component = 18 * sin(25) y component = 7.607 3. <16.313, 7.607>
Factor the polynomial and solve the inequality using a sign chart f(x) = 4x^3 - 7x^2 -21x + 18 >= 0
1.Synthetically divide to find the zeros x = 3, x = -2, x = 3/4 2. Plot on sign chart and test given conditions x = 3 (-)(+) == - x = 0 (-)(-) == + x = 1 (-)(+) == - x = 4 (+)(+) == + 3. f(x) >= 0: [-2, 3/4] U [3, inf)
The eighth grade class at a grade school has 9 girls and 14 boys, how many dates can be arranged?
126 dates can be created in the eighth grade class
Express the sum using summation notation. Use the lower limit of summation given and k for the index of summation: 2 + 4 + 6 + 8 + ... + 26
13Ek=1 (2k) = 182
long division of polynomials
1polynomial / 2polynomial 1p first term / 2p first term = thing on top multiply thing on top by 2p subtract result from original polynomial repeat with left over polynomial
To graph an ellipse equation you need
2 vertices 2 points on the minor axis Box connecting points and draw curvy lines
Evaluate 10C5
252
Difference of focal radii of a Hyperbola
2a
Major axis =
2a
Find the unique polynomial with real coefficents that meets these conditions: Degree = 4 Zeros: 1, -4, 2 - i; f(0) = -60
3x^4 - 3x^3 -33x^2 + 93x -60
Evaluate r*(v+w) r = <-1, 4, 1> v = <2, 2, 4> w = <-4, 7, 4>
46
Compute the exact value of the function for the given x-value f(x) = 5*2^x for x = 0
5
How many points are needed to graph a parabola?
5 1 vertex 2. Focal endpoints 2 random points found by plugging into equation
Find the sum of the following arithmetic sequence: -6, 0, 6, 12, 18
5Ek=1 (6k-12) = 30
Component form
< distance in the left/right direction, distance in the up/down direction >
What is vector j in component form?
<0, 1>
What is vector i in component form?
<1, 0>
Evaluate w/|w| w = <12m -3, -4>
<12/13, -3/13, -4/13>
Write the vector and parametric forms of the line through the point Po in the direction of v Po(5, -1, 3) v = <6, 8, -4>
<x, y, z> = <5, -1, 3> + t<6, 8, -4> x = 5 + 6t y = -1 + 8t z = 3 -4t
Vector equation:
<x,y,z> = <point vector> + t(directional vector>
Formula to Find magnitude given a vectors components
<x,y> sqrt( (x)^2 + (y)^2 )
Find the following matrices where A = [4 6] [2 5] B = [1 2] [3 2] A + B A - B -5A 4A + 2B
A + B = [5 8] [5 7] A - B = [3 4] [-1 3] -5A = [-20 -30] [-10 -25] 4A + 2B = [18 28] [14 24]
Write the statement as a power function equation: The area of a triangle varies directly as the square of the length s of its sides
A = k*s^2
Irredusible Quadratic
A quadratic that results in a imaginary number Use sqrt( b^2 -4ac) to determine if it is a irredusible quadratic
Reduced Row Echelon Form: Infinite Solutions
A row of zeros for one or more equations, and x and y must be written as variations of z and w.
Reduced Row Echelon Form: No solution
A row of zeros with a value at the end
An upper bound is represented by
All Non-negative numbers
A lower bound is represented by
Alternating non-negatives and non-positives
Systems of equations with different y-intercepts and the same slope...
Are parallel lines and don't intersect so no solution
The center of a 3 dimensional object is
C(h,k,L)
Find the center, verticies, and foci of the hyperbola (x+2)^2/144 - (y-6)^2/25 = 1
C: (-2, 6) V: (10, 6) and (-14,6) F: (11, 6) and (-15, 6)
Graph the hyperbola, and find its verticies, foci, and eccentricity 9(y-3)^2 - 4(x-1)^2 = 36
C: (1,3) V: (1, 5) and (1, 1) F: (1, 3 + sqrt(13) ) and (1, 3 - sqrt(13) ) e = sqrt(13) / 2 Reference Graph in Orange Notebook 8.3 Conic Sections Hyperbolas #9
Find the center, verticies, and foci of the hyperbola (y-2)^2/9 - (x-2)^2/4 = 1
C: (2,2) V: (2,5) and (2, -1) F: (2, 2+sqrt(13) ) and (2, 2 - sqrt(13) )
Determine the common ratio, the eigth term, a recursive rule, and an explicit rule for the geometric sequence: 1, -1, 1, -1, ...
Common Ratio = -1 a8 = -1 R: {a1 = -11, an+1 = an+8, n >= 1 E: 8n - 19
Find the common difference, the 10th term, a recursive rule, and an explicit rule for the nth term for the given sequence: 5, 14, 23, 32, ...
Common difference (d) = 9 a10 = 86 R: {a1 = 5, an+1 = an + 9, a >= 1 E: an = 9n - 4
Determine whether the infinite geometric series converges, if it does find its sum: 6 + 2 + 2/3 + 2/9 + 2/27 + ....
Convergent inf E k=1 (6 * (1/3)^k-1) = 9
Determine whether the infinite geometric series converges, if it does find its sum: 4 + 2 + 1 + 1/2 + 1/4 + ...
Converges inf E k=1 (4 *(1/2)^k-1) = 8
When given a magnitude and degree in bearings and asked for the component form ...
Convert bearing angle to a trig angle to find the component form: <mag*cos(trig angle) , mag*sin(trig angle)>
Odd multiplicty does what?
Crosses the x axis
Find the domain of the function f(x). Use limits to describe the end behavior at values of x not in the domain. f(x) = 1/x+7
D: (-inf, -7) U (-7, inf) lim x --> -7- f(x) = -inf lim x --> -7+ f(x) = inf
Identify the graph r = 5 + 4sinx
Dimpled limacon
limacon: 1 < a/b < 2
Dimpled limacon
Draw the inequality with its graph, indicate whether the boundary is included in or excluded from the graph y < x^2 + 5
Dotted parabola shifted 5 units up Shaded outside Reference System of Inequalities #9
Use the discriminant to determine whether the graph of the function is an ellipse/circel, a hyperbola, or a parabola: 5x^2 - 8xy + 5y^2 -21x + 22y =0
Ellipse
set of all points whose distance from 2 (foci) fixed points is a constant sum
Ellipse
vertical ellipses centered at (h,k)
Equation x-h^2/b^2 + y-k^2/a^2 = 1
Vertical things to know for ellipses centered at (0,0)
Equation x^2/b^2 + y^2/a^2 = 1 Vertex(0,+-a) Foci(0,+-c) Major axis is vertical
Tvertical parabola centered at (h,k)
Equation: (x-h)^2 = 4p(y-k) Directrix: y = # (p in the opposite direction from the center)
horizontal parabola centered at (h,k)
Equation: (y-k)^2 = 4p(x-h) Directrix: x = # (p in the opposite direction from the center)
horizontal ellipses centered at (h,k)
Equation: x-h^2/a^2 + y-k^2/b^2 = 1
vertical parabolas centered at (0,0)
Equation: x^2 = 4py Directrix: y = # (p in the opposite direction from the center)
Horizontal things to know for ellipses centered at (0,0)
Equation: x^2/a^2 + y^2/b^2 = 1 Vertex: (+- a, 0) Foci: (+-c, 0) Major axis is horixonatal
horizontal parabolas centered at (0,0)
Equation: y^2 = 4px Directrix: x = # (p in the opposite direction from the center)
End Behavior: Even Degree~_______ direction(s); Odd Degree~_______ direction(s)
Even = Same Odd = Opposite
How to find vertical asymptotes and their format
Find by setting the denominator = to 0 Once found make sure that the numerator is not 0 when plugging it in for x. If it does then it is not a VA VA: x = #
How to find x intercepts of rational functions and their format
Find x-int by setting the numerator = to 0 x-ints: x = #, x = #
How to find y intercepts of rational functions and their format
Find y-int by plugging 0 in for x y int: y = #
A function will look like a square root function if its exponent is ...
Greater than 0, but less than 1 0 < a < 1
If y^2 is on the left of a parabola it means
Horizontal
a under x^2 in a ellipse means
Horizontal
a under x^2 in a hyperbola means
Horizontal
In an exponential function there is 1 _____ but both ends ________ it
Horizontal Asymptote don't
Three things needed to describe the left and right hand side of a graph:
Increasing / Decreasing (read from left to right) Positive or Negative ( based on position on graph) Concave up / down (depends on the cave
If result of substitution/elimination is 0=0 the solution is ...
Infinite Solutions It is written as {(x,y) | one of the equations}
Is the following function exponential? If so, state the initial value and base. If not explain why y = 5^z
Is an exponential function inital value is 1 (a) base is 5
Use synthetic division to check that the number k is an upper bound for the real zeros of the function: F(x) = 6x^3 - 7x^2 + x - 6 k = 3
It is an upper bound all non-negative numbers
r^2 = a^2 * sin2x or r^2 = a^2 * cos2x
Lemniscate Type in calulator as r = sqrt(a^2 * cos2x) r= - sqrt(a^2 * cos2x)
Identify the graph theta = pi/3
Line; y = sqrt(3)x
Find the length of each pedal of the polar curve r = 2 + 4sin 2x
Long 6; Short 2 Take the absolute value of the minimum r value and maximum r value
When given a function to input into calculator to look at graph in parametric form...
Make sure in parametric mode and radians
Find the rectangular coordinate of (1.5 , 7pi/3)
Memorized angle denoms are 6,4,3 (3/4, 3sqrt(3)/4)
How many solutions does a system of equations with parallel lines have?
No Solutions
Is the following function exponential? If so, state the initial value and base. If not explain why y = z^9
Not an exponential function because the base is a variable
Find the order of the matrix. Indicate whether the matrix is square [2 1 -3]
Order: 1 x 3 Not square
Find the order of the matrix. Indicate whether the matrix is square [ 5 4 ] [-4 5 ]
Order: 2x2 Square
Determine the exponential function that satisfies the given conditions Initial Value: 0.6 halving every 5 days
P(t) = .6(.5)^t/5 If being halfed every ___ # of days divide the t variable by it
Determine the exponential function that satisfies the given conditions Initial Value: 16 decreasing at a rate of 60% per year
P(t) = 16(1-.60)^t P(t) = 16(.4)^t
Determine the exponential function that satisfies the given conditions Initial Value: 5 increasing at a rate of 18% per year
P(t) = 5(1+.18)^t P(t)=5(1.18)^t
Exponential Modeling and conditions
P(t) = P(1+r)^t r>0 exponential growth r<0 exponential decay r must be a decimal
Graph the system of inequalities {x^2 + y^2 <= 100 { y >= x^2
Parabala and circle Reference Linear Programming #3
Graph the equation: y^2 = -14x
Reference #4 Conic Section Parabolas: 8.1 Purple Notebook
Graph the equation: (x-4)^2/9 + (y-2)^2/4 = 1
Reference Circles and Ellipses Assignment 1 Purple Notebook #4
Use the formula for Sn to find the sum of the first 5 terms of the geometric sequence: 45, -9, 9/5, -9/25, ....
S5 = 4689/125
Arithmetic series formula
Sn = n/2(a1 + an) All are divided by 2
Elementary Row Operations
Swapping, Adding to Replace, and Dividing by an common factor
To find a common difference in a sequence ...
Take the second term minus the first term. Repeat to verify
If the equation is in the form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 you use what?
The Discriminant Test
Find rational zeros by using
The Rational Zero Therom gives you the possible rational zeros of a polynomial
Given C = piD; Write a description of the power function C is the circumference pi is a constant D is the diameter
The circumference C, is directly proportional to the diameter D, with a constant of variation pi
In a hyperbola a^2 is always what
The first denominator
Given f(x) = x^-1
The graph will look like a recipricoal functions
Given f(x) = x^-1/2
The graph will look like a recipricol function
In order to multiply two matrices
The number of columns of the first must = the number of row of the second n1 = m2
For parabolas if a < 0
The parabola opens downward
For parabolas if a>0
The parabola opens upward
Given f(x) = x^ 1/2
The shape of graph will be a square root function
Given f(x) = x^2
The shape of the graph will be a parabola
Direction of a scatterplot is based on
The way the dots head Going up = Positive direction Going down = Negative direction
Even multiplicity does what?
Touches the x axis
If the force and direction of motion are in the same direction then ...
W = |F| * |AB| |F| is how much weight , |AB| = how far Work = magnitude of the force * magnitude of the second vector
Does order matter in permutations
Yes
Use the factor therom to determine whether the first polynomial is a factor of the second polynomial x + 6 ; x^3 + 7x^2 + 4x - 12
Yes it is a factor
Find all of the zeros and write a linear factorization of the function: f(x) = x^3 - 11x^2 + 9x -99
Zeros: 11, +- 3i Factorization: (x-11)(x-3i)(x+3i)
Find a row echelon form from the given matrix [1 0 3 1] [2 2 4 6] [3 1 8 5]
[1 0 3 1] [0 1 -1 2] [0 0 0 0]
Find a row echelon form for the matrix [1 3 -1] [4 8 4] [-5 -8 -2]
[1 3 -1] [0 1 -2] [0 0 1]
Use the specified row transformation to change the matrix 7(R1) + R2 [1 3 9] [-7 4 -1] [2 7 0]
[1 3 9] [0 25 62] [2 7 0]
Write the system of equations as a matrix AX = B with A the coefficent matrix of the system {x + 7y = 19 {2x + 6y = 14
[1 7] * [x] = [19] [2 6] * [y] = [14]
Write the augmented matrix for the following system of equations {9x - y + 9z = -2 {3x - 2y + 7z = 1 {2y - 4z = -8
[9 -1 9 -2] [3 -2 7 1] [0 2 -4 -8]
limacon curve
a +- bcosx or a+-sinbx
Determine a<1 2>, a<2 1>, a<3 2> and a<1>*a<2 2> +3(a<2 3>) [ 4 12 8 ] [ 6 5 9 ]
a <1 2> = 12 a <2 1> = 6 a <3 2> = None 4 * 5 + 3(9) 20 + 27 47
The format x^2 + y^2 = # represents
a circle
If the function has an end behavior with a number then it implies that the function has ...
a horizontal asymptote
The fourth and seventh term of an arithmetic sequence are -2 and 16 respectively. Find the first term an a recursive rate for the nth term.
a1 = -20 R: {a1 = -20, an+1 = an+6, n>= 1
Find the first 6 terms and the 100th term of the sequence: an = n+2/n
a1 = 3 a2 = 2 a3 = 5/3 a4 = 3/2 a5 = 7/5 a6 = 4/3 a100 = 51/10
Find the first 4 terms and the 8th term of the recursively defined sequence: a1 = 5 an = an-1 - 4 n >= 2
a1 = 5 a2 = 1 a3 = -3 a4 = -7 a8 = -23
equation to find c in an ellipse
a^2 = b^2 + c^2
A sample space is
all possible outcomes of an event
Domain lists _____________
all possible x values
Range lists __________________
all possible y values
Quadratic Standard From
ax^2 + bx + c
1/b^# is the same as
b^-#
Evaluate the logarithimic expression: 6^log<6> 2
b^log<b> m = m 6^log<6> 2 = 2
Vertical asymptote end behavior rules
choose a number close to the asymptote on the left, and plug it in to the numerator to determine if it is positive or negative. Do the same to the denominator. Determine if the entire function is positive or negative with the number if positive = positive inf if negative = negative inf Repeat for right side
Identify the graph r = 3
circle, centered at (0,0) with a radius of 3
Sequences have
commas ,
Arithmetic sequences have
common differences (d)
Geometric Sequences have
common ratios (r)
Slope can be determined for vectors through their
component form
The total number of terms when a binomial is factored out is = to
degree + 1
Vectors
describe motion, not location Directed line segment Has direction and magnitutde
Identity matrixs are always
diagonal 1s filled in with 0s on the side
To find a common ratio
divide the second term by the first term. Repeat to verify
> or < get ____________ borders
dotted
If a >0 and 0<b<1 the function is an ______________ function add limits
exponential decay lim x-->-inf f(x) = inf lim x--> inf f(x) = 0
If a > 0 and b > 1 the function is an ________________ function add limits
exponential growth lim x-->-inf f(x) = 0 lim x --> inf f(x) = inf
A zero Function looks like
f(x) = 0
Exponetial function formula and conditions
f(x) = a * b^x a cannot = 0 b > 1
Basic Form of a Polynomial
f(x) = anX^n + an-xX^n-1 + ... a2X2 + aX + a
Power function formula
f(x) = k*x^a a and k can be any number except 0
Monomial Function formula
f(x) = k*x^n n is positive integers only or f(x) = k
Basic formula for a quadratic
f(x) = x^2
The factor therom
f(x) will have a factor of x-k if f(k) = 0
The counting principle with groups allows you to ...
find the number of combinations of groups of different objects
The number of complex zeros a function has is based on the
functions degree
A function will look like a parabola if it's exponent is ...
greater than 1 a>1
If the a^2 is under the x term in a hyperbola it is which direction
horizontal
Standard unit vectors
i = <1, 0, 0> j = <0, 1, 0> k = <0, 0, 1>
Two vectors are orthogonal ...
if and only if their dot product is 0
lim x --> #+ f(x) = -inf
implies that as x approaches a number from teh right side it goes towards negative infinity
lim x --> #- f(x) = inf
implies that as x approaches a number from the left side it goes towards infinity
If a function has >= or <= and two zeros in the same direction in a row you use
inf and -inf
A function will look like a recipricoal function if its exponent is
less than 0 a < 0
How is magnitude represented
like absolute value but with an arrow over the variable
Format of writing end behavior
lim x --> -inf / inf f(x) = inf / -inf
limacon with inner loop
limacon: 0 < a/b < 1
Dimpled limacon
limacon: 1 < a/b < 2
cartiod
limacon: a = b
The order of a matrix
m x n number of rows x number of columns
Midpoint for 3 dimensional points (x1, y1, z1) , (x2, y2, z2)
m( (x1+x2)/2, (y1+y2)/2, (z1+z2)/2)
Find the midpoint of the segment PQ P(-5, -1, 0) and Q(-3, 3, -2)
m(-4, 1, -1)
Expand the binoial using Pascal's triangle to find the coefficents (m+n)^3
m^3 + 3m^2n + 3mn^2 + n^3
A vector is a unit vector when the ...
magnitude of the vector = 1
When asked to match a polar equation with its graph ...
make sure you are in polar mode, and radians are set
as numbers approach a vertical asymptote from either side and go up the end behavior is ...
positive infinity
Polynomials must have _________ exponents, and can _________ decimals or fractions
positive integer; have no
In a three dimensional graph solid lines are
positive values
In a three dimensional graph every _________ must be drawn as a dashed line
possible combination of the points
Pascals triangle starts with
row 0
>= or <= get __________ borders
solid
i =
sqrt(-1)
When asked to find a certain term of (x+y)^# the choosing portion of the combination is
term - 1
If a power function is inversly proportional then ...
the constant of variation is divided by the variable k/ x^a
If a power function is directly proportional then ...
the constant of variation is multipliyed by the variable k * x^a
The exponent in a polynomial is called ...
the degree
If result of substitution/elimination is 2 constants that don't equal each other then
there is no solution
Subtraction of vectors v = <v1, v2, v3>, w = <w1, w2, w3>
v - w = <v1-w1, v2-w2, v3-w3>
Dot product of two vectors v = <v1, v2, v3>, w = <w1, w2, w3>
v*w = (v1*w1) + (v2*w2) + (v3*w3)
Calculate the expression using r = <-4, 3, -2> and w = <3, -7, 13> v*w
v*w = -59
If the a^2 is under the y term in a hyperbola it is which direction
vertical
Find the values of the variables w,x,y,z [w x] = [-2 5] [y z] [4 -2]
w = -2 x = 5 y = 4 z = -2
f(x) > 0 definition
when the function has y values greater than 0
f(x) >= 0
when the function has y values greater than or equal to 0
f(x) < 0
when the function has y values less than 0
Solve for x log x = 2
x = 10^2 x = 100
Going from polar coordinates to rectangular coordinates...
x = r * cos(x) y = r* sin(x) 1. If an known angle measurement, use an exact value.
Identify if the graph has symmetry r = 4-3cosx
x axis symmetry
Identify if the graph has symmetry r = 5cos2x
x axis symmetry y axis symmetry orgin symmetry of 180 degrees
Find the intercepts, vertical asymptotes, and the end behavior asymptote and graph the function together with its end behavior asymptotes f(x) = x^4 + 3 / x + 1
x-int: None y-int: y = 3 VA: x = -1 lim x --> -1 - f(x) = -inf lim x --> -1 + f(x) = inf HA: None EBA: x^3 - x^2 + x -1 Refer to graph on Asymptotes and Intercepts #14
Graph the function f(x) = x+8 / x^2 -4x -32 Find all asymptotes. List x and y intercepts
x-int: x = -8 y-int: y = -1/4 VA: x = -4, x= 8 lim x --> -4 - f(x) = inf lim x --> -4 + f(x) = -inf lim x --> 8 - f(x) = -inf lim x --> 8 + f(x) = inf HA: y = 0 lim x --> -inf f(x) = 0 lim x --> inf f(x) = 0 Refer to graph on Asymptotes and Intercepts # 3
Write the polynomial in standard form, and identify the zeros of the function and the x-ints of the graph f(x) = (x - 9i)(x + 9i)
x^2 + 81 --> standard form zeros: +/- 9i (no x intercepts because imiginary numbers)
Find the equation of the parabola with a focus at (0,12) and directrix y = -12
x^2 = 48y
equation for a horizonal hyperbola centered at (0,0)
x^2/a^2 - y^2/b^2 = 1
equation for vertical hyperbola asymptotes centered at (0,0)
y = +- a/b x
equation for horizontal hyperbola asymptotes centered at (0,0)
y = +- b/a x
Use a graphing calculator in function mode to graph the hyperbola: x^2/9 - y^2/64 = 1
y = +- sqrt(64x^2-576)/3
Vertex form
y = a(x-h)^2 + k A is Leading Coefficent in standard form h is = -b/2a k is y in standard form when x is pluged in
Identify if the graph has symmetry r = 3+3sinx
y axis symmetry
Asymptote of a vertical hyperbola centered at (h,k)
y-k = +- a/b(x-h)
Asymptote of a horizontal hyperbola centered at (h,k)
y-k = +- b/a(x-h)
Write an equation for the parabola with a vertex at the orgin and focus (7,0)
y^2 = 28x
Find an equation in standard form for the hyperbola that satisfies the given conditions: Foci (0, +- 10) Transverse Axis length = 4
y^2/4 - x^2/96 = 1
Find an equation in standard form for the hyperbola that satisfies the given conditions: Center (0,0) b = 7 e = 25/24 Vertical Focal axis
y^2/576 - x^2/49 = 1
equation for a vertical hyperbola centered at (0,0)
y^2/a^2 - x^2/b^2 = 1
Irrational zeros are
zeros not on the list
Recursive setup for an arithmetic sequence
{ an = #; an+1 = an + d; n >= 1
Write the matrix equation as a system of equations [3 -3] * [x] = [-3] [2 4] * [y] = [3]
{3x - 3y = -3 {2x + 4y = 3
Write the system of equations that corresponds to the following augmented matrix [3 -2 3] [4 6 -3]
{3x -2y = 3 {4x + 6y = -3
Determine if the function is a power function, if it is then state the power and the constant of variation. If not explain why. f(x) = k / x^2
Power Function Power: -2 Constant: K
When writing out a linear factorization with just factors ...
make sure to include an a value that causes the y-intercepts to be comparable
In a three dimensional graph the x coordinate must be
measured with a ruler and spaced accordingly
To multiply a matrix by a scalar simply
multiply each position in the matrix by the scalar
Combination equation
nCr n!/(n-r)! * r!
Summation notation
nEk=1 (ak) n = the number of terms k = 1 is the counter variable (ak) = the original explicit formula
Permutation equation
nPr n!/(n-r)! n = # of available items r = # of chosen items
Given a certain number of choices for a list of items the equation is
n^c where n is the number of choices and c is the number of questions
Solve the system of equations by finding the reduced row echelon form for the augmented matrix {x + y - 3z = 1 {x - z - w = 3 {2x + y - 4z - w = 4
{(z+w+3, 2z-w-2, z, w)}
Focal width of a parabola =
| 4p |
Magnitude of a vector (v) v = <v1, v2, v3>, w = <w1, w2, w3>
|v| = sqrt( v1^2 + v2^2 + v3^2)
In an expierement on social interaction, 3 people will sit in 3 seats in a row how many ways can this be done?
There are 6 ways that the individuals can sit
A multiple choice test consists of 7 questions with each equation having 5 possible outcomes. How many different ways are there to make the answers
There are 78,125 different ways to make the answers 5^7
Find an equation in standard form for the hyperbola that satisfies the given conditions: Center (0,0) a = 3 e = 4 Horizontal Focal axis
x^2/9 - y^2/135 = 1
synthetic division of polynomials
zero of divisor | term1 term 2 term 3 1. Bring down 1 2. Multiply by zero 3. Add to term 2 4. Repeat 5. Last number is the remainder
Solve the system of equations by finding the reduced row echelon form of the augmented matrix {-2x + 2y -z = 4 {-4x -2y + z = 8
{(-2, 1/2z, z)}
Solve the system of equations by finding the reduced row echelon form of the augmented matrix {x + y - z = 2 {4x - y + z = -2 {x - 3y + 2z = -23
{(0, 19, 17)}
Solve the linear systems of equations by computing X = A^-1B {x + 5y + 5z = -19 {x + 4y + 5z = -16 {x + 5y + 4z = -16
{(11, -3, -3)}
Solve the system of equations by finding a row echelon form for the augmented matrix {x -y + 2z = 7 {2x + 3y -6z = -11 {-5x + 2y -3z = -19
{(2, -3, 1)}
Using only algebra, find a cubic function with the given zeros: sqrt(3) , -sqrt(3), 5
1. Write as factors (x - sqrt(3) ) (x + sqrt(3) ) (x - 5) 2. Multiply x^3 - 5x^2 - 3x + 15
Describe how the graph of f(x) = 3x-4/x + 2 can be obtained by transforming the graph of the recipricol function g(x) = 1/x. Identify the horizontal and vertical asymototes and use limits to describe the cooresponding behavior. Sketch the graph of the function
1. Write as vertex form synthetically divide the numerator by denominator 3 + -10/x+ 2 --> -10(1/x+2) + 3 From 1/x left 2 units, followed by a vertical stretch by a factor of 10, then a reflection across the x-axis, and then up 3 units 2. Identify asymptotes and their limits VA: x = -2 HA: y = 3 lim x --> -2 - f(x) = inf lim x --> -2 + f(x) = -inf lim x --> -inf f(x) = 3 lim x --> inf f(x) = 3 3. Sketch Refer to graphs of rational functions #7
Describe how the graph of f(x) = 3x-1/x+1 can be obtained by transforming the graph of the recipricol function g(x) = 1/x. Identify the horizontal and vertical asymototes and use limits to describe the cooresponding behavior. Sketch the graph of the function
1. Write as vertex form synthetically divide the numerator by denominator 3 - 4/x+ 1 --> -4(1/x+1) + 3 From 1/x left 1 units, followed by a vertical stretch by a factor of 4, then a reflection across the x-axis, and then up 3 units 2. Identify asymptotes and their limits VA: x = -1 HA: y = 3 lim x --> -1 - f(x) = inf lim x --> -1 + f(x) = -inf lim x --> -inf f(x) = 3 lim x --> inf f(x) = 3 3. Sketch Refer to graphs of rational functions #8
Determine the formula for the exponential whose values are given (0,5/4) and (1,5/16)
1. Write formula y = a * b^x 2. Plug in the point of the y-intercept and solve for a 5/4 = a *b^0 (b^0 is 1) 5/4 = a 3. Solve for b using the other point and a plugged in 5/16 = 5/4 * b^1 1/4 = b^1 b = 1/4 4.Write as the function f(x) = 5/4 * 1/4^x
Determine the formula for the exponential whose values are given (0,7) and (2,14)
1. Write formula y = a * b^x 2. Plug in the point of the y-intercept and solve for a 7 = a *b^0 (b^0 is 1) a = 7 3. Solve for b using the other point and a plugged in 14 = 7 * b^2 2 = b^2 b = sqrt(2) 4.Write as the function f(x) = 7 * 2^x/2
A rectangle has an area of 144in^2 and a perimeter of 60 in. Find the dimensions of the rectnagle
1. Write the basic equations 2w + 2L = P w * L = A 2. Plug in Values 2w + 2L = 60 w * L = 144 3. Solve for a variable 2L = 60 -2w L = 30 - w 4. Subsitiute w * (30 - w) = 144 30w - w^2 = 144 w^2 - 30w + 144 (w-6)(w -24) w = 6, w = 24 5 Plug in value to other equation and solve for other variable L = 30 - 6 L = 24 6. Write the dimensions as an ordered pair (6,24)
For the following, find function f(x) defined by a polynomial of degree 3 with real coefficents that satisfy the given conditions: Zeros: -3, -1, 4 LC: 6
1. Write zeros as factors (x +3) (x + 1) ( x- $) 2. Multiply and simplify x^3 - 13x -12 3. Apply conditions 6(x^3 - 13x -12) 6x^3 -78x - 72
Use the Rational Zero Therom to write a list of all potential rational zeros. Then determine which ones if any are real zeros f(x) = x^3 - 6x^2 - 9x + 14
1. p/q --> 14/1 1 2 7 14 -------- 1 possible rational zeros +/- 1, 2, 7, 14 actual zeros are 1, 7, -2
Sum of Focal Radii =
2a
The Conjugate axis =
2b
The Focal Axis =
2c
Use synthetic division to perform each division 2x^3 + 10x^2 + 8c - 17 ----------------------- c + 3
2x^2 + 4x - 4 - 5/c+3 -5/c+3 is the remainder
Identify max r value r = 3cos3x
3
Given r = 3 sin3x How many pedals are there?
3 Odd "b" in a*cosbx is the number of pedals
Basic Domian signs with solving inequalities
<= or >= will usually have a [ and a ) < or > will have ( and )
Use Substitution to solve the system of linear equations. Check your Solutions {x + y - 5z = 7 {3y + z = 11 {z = -1
3y + -1 = 11 y = 4 x + 4 -5(-1) x = -2 {(-2, 4, -1)}
Given r = 3 cos2x How many pedals are there?
4 Even number "b" in a*cosbx is 2 times the number of pedals
In a three dimensional graph how is the y line drawn left to right
Dashed --> Vertex ---> Solid
How to find horizontal asymptotes and their format
Find by comparing the degrees of the numerator and denominator If numerator < denominator then y = 0 is a HA If numerator = denominator then y = LC num / LC denom If numerator > denominator then No Horizontal Asymptote, but need to look for End Behavior Asymptote HA: y = #
Systems of equations with the same equation...
Have infinitely many solutions
Find the verticies and foci of the hyperbola: x^2/9 - y^2/8 = 1
Vertices: (3,0) (-3,0) Foci: (sqrt(17), 0) (-sqrt(17), 0)
Find the verticies and foci of the hyperbola: 9x^2 - 4y^2 = 36
Verticies: (2,0) (-2,0) Foci: (sqrt(13), 0) (-sqrt(13), 0)
Basic Form of a Polynomial Function and rules
ax^n + bx^n-1 + cx^n-2 ... + yx + z - a can never be 0 -coefficents must be real numbers
Complex zeros are always found in
conjugate pairs 2 irrational 1 rational or 2 irrattional 2 rational etc.
k is the
constant of variation/proportion
Identify the graph r = 5 + 2 cosx
convex limacon
The angle between two vectors is
cos-1( u*v / |u| * |v| )
Solve the linear systems of equations by computing X = A^-1B {x + 4y = -24 {6x + 5y = -49
{(-4, -5)}
The recipricol function
1/x
the set of all points whose distance's from two fixed points (Foci) have a common difference
Hyperbola
Given f(x) = x^4.6
The shape of the graph will be a parabola
limacon: a/b >= 2
Convex limacon
What is multiplicity?
Multiplicity is the exponent of a factor
A list of numbers seperated by commas that can be finite or infinte
Sequence
Find an equation in standard form for the ellipse that satisfies the given conditions: Major axis endpoints (-8,0) and (-8,-6) Minor axis length of 4
(x+8)^2/4 + (y+3)^2/9 = 1
Find an equation in standard form for the hyperbola that satisfies the given conditions: Transverse Axis endpoints (4,5) and (4, -1) Conjugate axis length = 8
(y-2)^2/9 - (x-4)^2/6 = 1
Find teh equation in standard form of the parabola described: Vertex at (-2, 3); Focus at (-1,3)
(y-3)^2 = 4(x+2)
equation for a vertical hyperbola centered at any given point
(y-k)^2/a^2 - (x-h)^2/b^2 = 1
Determine the x value that causes the polynomial to a) be zero b) be positive c) be negative f(x) = (5x^2 + 9)(x-6)^2(x+2)^3
*Note: b/c 5x^2 +9 is squared inside the parentheses and squares can only result in non-negative numbers, there is no zero for this factor a) Zeros: x = 6, x = -2 b) f(x) > 0: (-2, 6) U (6, inf) c) f(x) < 0: (-inf, -2) 0 0 <------------|------------------------|-----------------> -2 6 Test conditions x = -3 (+)(+)(-) == - x = 0 (+)(+)(+) == + x = 7 (+)(+)(+) == +
Find if possible AB and BA A = [-2 3] [-2 1] [1 5] B = [-4 -5 0] [1 -1 -1]
1. Compare orders for A * B 3x2 2x3 (Rows and Columns the same) 2. Mulitply AB = [-2(-4)+3(1) -2(-5)+3(-1) -2(0)+3(-1)] [-2(-4)+1(1) -2(-5)+1(-1) -2(0)+1(-1)] [1(-4)+5(1) 1(-5)+5(-1) 1(0)+5(-1)] AB = [11 7 -3] [9 9 -1] [1 -10 -5] 3. Compare orders for B*A 2x3 3x2(Rows and Columns the same) 4. Multiply BA = [-4(-2)+-5(-2)+0(1) -4(3)+-5(1)+0(5)] [1(-2)+-1(-2)+-1(1) 1(3)+-1(1)+-1(5)] BA = [18 -17] [-1 -3]
Find if possible AB and BA A = [8 6] [-1 9] B = [-3 -8] [-8 0]
1. Compare the orders for A * B 2x2 2x2 (Rows and columns match) 2. Multiplty AB = [8(-3)+6(-8) 8(-8)+6(0)] [-1(-3)+9(-8) -1(-8)+9(0)] AB = [-72 -64] [-69 8 ] 3. Compare the orders for B * A 2x2 2x2 (Rows and columns match 4. Multiply BA = [-3(8)+-8(-1) -3(6)+-8(9)] [-8(8)+0(-1) -8(6)+0(9)] BA = [-16 -90] [-64 -48]
When given a magnitude and degree in bearings of two vectors and asked for the combined magnitude and direction angle ...
1. Convert bearing angle to a trig angle to find the component form for both vectors: <mag*cos(trig angle) , mag*sin(trig angle)> 2. Add the the two component forms to find the component form of the resultant vector 3. Find the magnitude of the resultant 4. Find the angle inside the triangle formed by the componets and the resultant 5. Find the bearing for the direction angle
Find a unit vector in the direction of the given vector w = -i - 2j
1. Convert i's and j's to component form w= -1<1,0> - 2<0,1> W = <-1, 0> - <0, 2> W = <-1, -2> 2. Determine the magnitude of the vector |w| = sqrt( (-1)^2 + (-2)^2 ) |w| = sqrt(5) 3.Using the magnitude determine the unit vector unit vector = 1/sqrt(5) * <-1, -2> unit vector = <-1/sqrt(5), -2/sqrt(5)> unit vector = < - sqrt(5) / 5, -2sqrt(5) / 5>
Sketch the polynomial function graph for the given zeros and multiplicities -4 multiplicity of 3 3 multiplicity of 2
1. Convert zero to factor form (x+4)^3 (x-3)^2 2. Odd multiplicities = cross x axis 3. Even multiplicites = touch x axis
Find a polynomial with real coefficents that has the given zeros: -1, 5-4i
1. Convert zeros to factors (x + 1) (x - (5 - 4i) ) (x - (5 + 4i) ) (X + 1) (x -5 +4i) (x -5 -4i) 2. Multiply and Simplify x^3-9x^2+31x+41
Write a polynomial function minimum degree in standard form with real coefficents whose zeros include those listed zeros: 3i and -3i
1. Convert zeros to factors (x - 3i) (x + 3i) 2. Multiply and Simplify x^2 + 9
Solve: (3/x-1) + x = 9
1. Identify LCD --> x-1 2. Multiply each term by LCD 3 + x(x-1) = 9(x-1) 3. Get all terms on 1 side x^2 - 10x + 12 =0 4. Solve for the zeros x = -(-10) +- sqrt((-10)^2 - (4 * 1 * 12) ---------------------------------- 2(1) x = 10 +- sqrt(52) ------------- 2 x= 5 +- sqr(13) 5. Check for extraneous solutions Both are Solutions {5+ sqrt(13), 5-sqrt(13)}
Determine wheter the vectors u and v are parallel, orthogonal, or neither u = <2, -7> v = <-4, 14>
1. Determine Slope slope of u = (-7/2) slope of v = (14/-4) --> (-7/2) Same slopes = Parallel Vectors
Determine wheter the vectors u and v are parallel, orthogonal, or neither u = <5, -6> v = <-12, -10>
1. Determine Slope slope of u = (-6/5) slope of v = (-10/-12) --> (5/6) 2. Determine if orthognal by finding the dot product u*v = 0 They are Orthogonal Vectors
Let vector U = <-1, 3> and vector V = <2,4> Find the component form of the vector 2u - 4v
1. Determine the component form of vector 2u and vector -4v 2u = 2 * <-1, 3> --> <-2, 6> -4v = -4 * <2, 4> --> <-8, -16> 2. Using vector arithmitic add the first terms and the second terms <-2,6> + <-8, -16> --> <-10, -10> Vector 2u - 4v = <-10, -10>
Find a unit vector in the direction of the given vector u = <-2, 4>
1. Determine the magnitude of the vector |u| = sqrt( (-2)^2 + (4)^2 ) |u| = sqrt(20) |u| = 2sqrt(5) 2. Using the magnitude determine the unit vector unit vector = 1/s2qrt(5) * <-2, 4> unit vector = <-2/2sqrt(5), 4/2sqrt(5)> unit vector = <-1/sqrt(5), 2/sqrt(5)> unit vector = <-sqrt(5)/5, 2sqrt(5)/5>
Find the exact solution algebriacally 2 * 2^x/3 = 32
1. Divide 2^x/3 = 16 2. Find out what 2 raised to = 16 2^4 = 16 2^x/3 = 2^4 x/3 = 4 x = 12 {12}
Graph (calculator) the polynomial function, and locate its extrema and zeros: f(x) = -x^4 + 13x
1. Draw graph with viewing window 2. x = 0 x = cubrt(13) -- 2.351
Plotting points with given polar coordinate (-1 , 2pi/5)
1. Draw picture and convert 2pi/5 to degree 2pi/5 = 72 degrees 1rst quadrant 2. Then draw from magnitude 0 backwards -1 Dot is placed in 3rd quadrant because magnitude is negative
Solve by the elimination method {2x - 5y = 33 {3x + 6y = 9
1. Eliminate 1 variable 3(2x - 5y) = 3(33) -2(3x + 6y) = -2(9) 6x - 15y = 99 -6x -12y = -18 -27y = 81 2. Solve for that variable y = 81/-27 y = -3 3. Plug in variable for 1 of the equations and solve for other variable 3x + 6(-3) = 9 3x -18 = 9 3x = 27 x = 9 4. Check by plugging values into other equation 2(9) - 5(-3) = 33 18 + 15 = 33 33 = 33 --> True 5. Write as an ordered pair {(9, -3)}
Solve by the elimination method {5x - 15y = 2 {-20x + 60y = -12
1. Eliminate 1 variable 4(5x - 15y) = 4(2) 20x -60y = 8 -20x + 60y = -12 0 = -4 --> False No Solution
State the values for constant of variation and the exponent Describe the right hand side Determine if function is odd or even Describe the left hand side if any f(x) = 1/2x^-3
1. Exponent = -3; Coefficent = 1/2 2. Right Hand Side: -Decreasing -Positive -Concave Up 3. Odd function 4. Left Hand side: -Decreasing -Negative -Concave Down
State the values for constant of variation and the exponent Describe the right hand side Determine if function is odd or even Describe the left hand side if any f(x) = 3x^1/4
1. Exponent = 1/4 ; Constant = 3 2. Right Hand Side -Increasing -Positive -Concave Down 3. Even root = Undefined 4. No left hand side
State the values for constant of variation and the exponent Describe the right hand side Determine if function is odd or even Describe the left hand side if any f(x) = -2x^4/3
1. Exponent = 4/3; Constant = -2 2.Right Hand Side -Decreasing -Negative -Concave Down 3. Even function 4. Left Hand Side -Increasing -Negative -Concave Down
Rewrite the equation in vertex form: f(x) = 3x^2 + 5x - 4
1. Find h using -b/2a h = -5/2(3) h = -5/6 2. Find k by plugging h into standard form f(-5/6) = 3(-5/6)^2 + 5(-5/6) -4 f(-5/6) = -73/12 3. Identify a as leading coefficent a = 3 3. Plug into Vertex form y = 3(x + 5/6)^2 -73/12
Prove that the vectors u and v are orthogonal u = <2, 3> v = < 3/2, -1>
1. Find if the dot product of the two vectors is 0 u*v = 2* 3/2 + 3 * -1 u*v = 3 + -3 u*v = 0
Find the magnitude and direction angle of the vector <-1,2>
1. Find magnitude sqrt( (-1)^2 + (2)^2 ) mag: sqrt(5) 2. Find the direction angle Find the angle inside the triangle using trig angles tan -1 ( 2/1) a = 63.4 degrees Identify the Quadrant 2nd Quadrant = theta = 180 - a Calculate Theta = 180 - 63.4 theta = 116.6 degrees 3. State magnitude and direction angle sqrt(5) @ 116.6 degrees
Find the magnitude and direction angle of the vector <3,4>
1. Find magnitude sqrt( (3)^2 + (4)^2 ) mag: 5 2. Find the direction angle <3, 4> is over 3, up 4 so it would be in the first quadrant and have an x value of 3 and a y value of 4 Using this knowledge find the angle of the triangle formed. tan-1 (4/3) = 53.1 degrees 4. 5 @ 53.1 degrees
Tell whether the function is an exponential growth or decay and find the constant rate of crowth /decay f(x) = 4.8 * 1.05^x
1. Find r 1.05 = 1 +r r = .05 2.Compare r r>0 = true so function is expoential growth
Find the parametrization for the curve The line through the points (-2,5) and (4,2)
1. Find slope vector -2 --> 4 = 6 5 --> 2 = -3 <6, -3> 2. Plug into parametrization formula <x,y> = slope vector * t + a starting point <x,y> = <6,-3> * t + (4,2) 3. Assign x and y values x = 6t + 4 y = -3t + 2
State the domain of the function of f(x) = log[x(x+7)]
1. Find the Vertical asymptote x= 0 x = -7 2. Place into a sign chart + -7 - 0 + 3. Find the x intercepts: x(x+7) = 1 x^2 + 7x -1 = 0 quadratic formula x = 0.140 x = -7.140 4. State the domain (-inf , -7) U (0, inf) 5. Choose two random points to the left and right, and then plot them and draw curves
How to solve polynomial inequalities
1. Find the Zeros and Undefined Values 2. Place Zeros on the sign chart 3. Test the "Regions"
Find vectorv with the given magnitude and same direction as u |v| = 2 u = <3, -3>
1. Find the angle of the triangle tan -1 (3/3) = 45 degrees 2. Use triangle angle to determine trig angle Quadrant 4: 360 - 45 = 315 3. Find the component form of vector v < 2 * cos(315) , 2* sin(315) > <1.414, -1.414>
Let Q = (3,4) and S = (2, -8) Find the component form and magnitude of 2QS
1. Find the component form of 2QS by finding the distance in the x and distance in the y 2QS = 2 * <-1, -12> 2QS = <-2, -24> 2. Find the magnitude of 2QS using the component form |2QS| = sqrt( (-2)^2 + (-24)^2 ) |2QS| = sqrt(580) |2QS| = 2sqrt(145) Component form of 2QS = <-2, -24> Magnitude of 2QS = 2sqrt(145)
Let P = (-2,2) amd Q = (3,4) Find the component form and magnitude of vector PQ
1. Find the component form of PQ by finding the distance in the x and distance in the y PQ = <5,2> 2. Find the magnitude of PQ using the component form |PQ| = sqrt( (5)^2 + (2)^2 ) |PQ| = sqrt(29) Component form of PQ = <5,2> Magnitude of PQ = sqrt(29)
Prove that vector RS and vector PQ are equivilant by showing that they represent the same vector R=(2,1) S = (0,-1) P = (1,4) Q=(-1, 2)
1. Find the component form of RS by finding the distance in the x and distance in the y RS = <2,1> --> <0,-1> RS = <-2, -2> 2. Find the component form of PQ by finding the distance in the x and distance in the y PQ = <1,4> --> <-1,2> PQ = <-2,-2> 3. Compare RS = PQ <-2,-2> = <-2,-2>
Find the angles between the vectors u = <-4, -3> v = <-1, 5>
1. Find the dot product u*v = -11 2. Find the magnitude of each vector |u| = sqrt( (-4)^2 + (-3)^2 ) |u| = 5 |v| = sqrt( (-1)^2 + (5)^2 ) |v| = sqrt(26) 3. Plug in magnitude and dot product into formula theta = cos -1 (-11 / 5 * sqrt(26) ) theta = 115.6
Find the vector projection of u onto v. Then write u as a sum of two orthogonal vectors one of which is projection of u onto v. u = <-8, 3> v = <-6, -2>
1. Find the dot product u*v = 42 2. Find the magnitude of v |v| = sqrt( (-6)^2 + (-2)^2 ) |v| = sqrt(40) 3. Plug into the formula ProjvU = (42 / sqrt(40)^2) * <-6, -2> ProjvU = <-6.3, -2.1> What will make the dot product 0 and = to u -8 - -6.3 = -1.7 3 - -2.1 = 5.1 u = <-6.3, -2.1> + <-1.7, 5.1>
Write an equation for the linear function f satisfying the given conditions f(-4) = 6 and f(-1) = 2
1. Find the slope of the function by finding each point Point 1 (-4, 6) Point 2 (-1, 2) Slope = 2 - 6 / -1 - -4 Slope = -4 / 3 2. Using the slope and slope intercept form and a given point, find y intercept y = mx + b 2 = -4/3(-1) +b 2 = 4/3 + b 2/3 = b 3. Plug into slope intercept form y = -4/3x + 2/3
Write an equation for the linear function f satisfying the given conditions f(-5) = -1 and f(2) = 4
1. Find the slope of the function by finding each point Point 1 (-5, -1) Point 2 (2, 4) Slope = 4- -1 / 2 - - 5 Slope = 5/7 2. Using the slope and slope intercept form, find y intercept y = mx + b 4 = 5/7(2) +b 4 = 10/7 + b 18/7 = b 3. Plug into slope intercept form y = 5/7x + 18/7
Write an equation for the linear function f satisfying the given conditions f(0) = 3 and f(3) = 0
1. Find the slope of the function by finding each point Point 1 (0,3) Point 2 (3,0) Slope = 0 - 3 / 3 - 0 Slope = -1 2. Using the slope and slope intercept form and a given point, find y intercept y = mx + b 0 = -1(3) +b 0 = -3 + b 3 = b 3. Plug into slope intercept form y = -x + 3
Find the x intercept A and the y intercept B and the coordinates of the point P so that AP is perpendicular to the line and |AP| = 1 Given 3x - 4y = 12
1. Find the x and y intercepts x intercept is found when y = 0 3x - 4(0) = 12 3x = 12 x = 4 A(4,0) y intercept if found when x = 0 3(0) - 4y = 12 -4y = 12 y = -3 B(0,-3) AB<-4,-3> ; Slope = -3/-4 = 3/4 perpendicular slope = -4/3 2.From slope determine point P (3, -4) or (-3,4) 3. Find the magnitude of point P sqrt( (3)^2 + (-4)^2 ) = 5 4. Find the unit vector of point P because it asks for the magnitude of AP to be 1 (3, -4) * 1/5 and (-3,4) * 1/5 (.06, -0.8) and (-0.6, 0.8) 5.Add to A (4,0) P = (4.6, -.8) or (3.4, 0.8)
Determine the real values of x that can cause the function to be a) Zero, b) Undefined, c) Positive, d) Negative f(x) = x sqrt(x+10)
1. Find the zeros (What makes the numerator 0) x = 0 sqrt(x+10) = 0 x = -10 2. Undefined (Negative roots, or 0 denominator) x < -10 3. Place values on sign chart and test regions Cross out anything behind -10 4. Determine what makes the function positive f(x) > 0: (0, inf) 5. Determine what makes the function negative f(x) < 0: (-10, 0)
Find the x intercept A and the y intercept B and the coordinates of the point P so that AP is perpendicular to the line and |AP| = 1 Given 3x-7y = 21
1. Find x and y intercepts x intercept is when y = 0 3x - 7(0) = 21 3x = 21 x = 7 A (7,0) y intercept is when x = 0 3(0) - 7y = 21 -7y = 21 y = -3 B(0, -3) 2. Use perpendicular slope to determine possible points AB = <-7, -3> slope = -3/-7 = 3/7 perpendicular slope = -7/3 Possible points: (-3, 7) or (3, -7) 3. Find unit vector of possible points magnitude = sqrt(58) (-3, 7) * 1/sqrt(58) or (3,-7) * 1/sqrt(58) 4. Add on to cooridate A (-0.394, 0.919) + (7,0) --> (6.606, .919) or (.394, -.919) + (7,0) --> (7.394, -0.919)
Graph the rational function with its graph 4 + 1/x+1
1. Find x-intercepts by converting the function into 1 function 4(x+1)/ x+1 + 1/(x+1) = 4(x+1) + 1 / (x+1) 4x + 4 + 1 / (x+1) 4x + 5 / x + 1 ; x-int: x = -5/4 2. Find y intercept Plug 0 in for x y-int = 5 3. Find VA VA: x = -1 lim x --> -1 - f(x) = -inf lim x --> -1 + f(x) = inf 4. Find HA (Compare degree of leading terms) num = denom --> 0 / 1 = 0 + 4 --> 4 HA: y =4 Refer to graph on Asymptotes and Intercepts #9
Using only algebra, find the cubic function with the given table of values. x = -4 , 0 , 3 , 4 y = 0, 96, 0, 0
1. Find zeros where y = 0 (x + 4) (x - 3) (x-4) 2. Multiply and simplify x^3 - 3x^2 - 16x + 48 3. Using table we know yint is 96, satisfy condition by multiplying by 2 2x^3 - 6x^2 - 32x + 96
Solve by the substitution method { x - 2y = 7 {3x - 6y = 15
1. Get a variable by itself x - 2y = 7 x = 2y + 7 2. Substitute and solve for other variable 3(2y+7) - 6y = 15 6y + 21 - 6y = 15 21 = 15 --> 2 constants that != --> No Solution
Solve by the substitution method { 6x + y = 38 {x - 3y = 19
1. Get a variable by itself x - 3y = 19 --> x = 3y + 19 2. Substitute and solve for other variable 6(3y + 19) + y = 38 18y + 114 + y = 38 19y + 114 = 38 19y = -76 y = -4 3. Plug answer back into orginal equation x = 3(-4) + 19 x = 7 4. Check answer with other equation 6(7) + -4 = 38 42 + -4 = 38 38 = 38 --> True 5. Write as ordered pair {(7, -4)}
Using only algebra, find a cubic function with the given zeros: -1 , 2 , -6
1. Write as factors (x + 1) ( x - 2) ( x + 6) 2. Multiply x^3 + 5x^2 -8x - 12
Solve {x^2 + y^2 = 25 {x + y = 4
1. Get circle = y y^2 = 25 - x^2 y = +-sqrt(25 - x^2) 2. Substitute for other equation and solve x + y = 4 y = -x + 4 sqrt(25 - x^2) = -x +4 25 - x^2 = (-x +4) (-x + 4) 25 - x^2 = x^2 - 8x +16 2x^2 -8x -9 = 0 x = -(-8) +- sqrt( (-8)^2 - 4(2)(-9) ----------------------------------- 2(2) x = 4.92 , x = -0.92 3. Solve for other variable y = -(4.92) + 4 --> y = -0.92 y = -(-0.92) + 4 --> y = 4.92 4. Write as ordered pair {(-0.92, 4.92) , (4.92, -0.92)}
Determine the number of solutions of the system {2x - 4y = 10 {3x - 6y = 15
1. Get equations to slope form 2x - 4y = 10 -4y = -2x + 10 y = 1/2x -5/2 3x - 6y = 15 -6y = -3x + 15 y = 1/2x -5/2 2. Compare slope and y-intercepts y = 1/2x - 5/2 y = 1/2x - 5/2 3. Same equation = Infinitely Many Solutions
A 2000 lb car is parked on a street that makes an angle of 12 degrees with the horizontal. Find the the force required to keep the car from sliding down the hill and the force perpendicular to the street.
1. Gravational force mag * sin(x) 2000 * sin(12) F = 415.823 2. Horizontal component mag * cos(12) 1956.295
It takes 9 units of carbs and 5 units of protien to satisfy minimum weekly requirements. The meat consists of 2 units of carbs and 2 units of protien per pound. The cheese contains 3 units of carbs and 1 unit of protien per pound. The meat costs $3.50 per lb and the cheese costs $4.80 per lb. Find out how many lbs of each are needed to meet requirements at the lowest cost.
1. Idenify Constraints M >= 0 (Cannot have negative meat) C >= 0 (Cannot have negative cheese) carbs: 2M + 3C >= 9 (Think of what each item has) protien: 2M + C >= 5 (Think of what each item has) P = 3.50M + 4.80C 2. Graph, and identify corners, and what they produce with the objective function(price) Corners (M,C): (0, 5) --> 24 (1.50, 2) --> 14.85 (4.5, 0) --> 15.75 3. Identify minimum 1.50 lbs of meat and 2 lbs of cheese at $14.85
Solve the inequality 1/x+4 + 1/x-8 <= 0
1. Identify LCD (x-8)(x+4) 2. Multiply each term by LCD 3. Zeros x = 2 4. Undefined x = 8, x = -4 5. Sign Chart 6. f(x) <= 0: (-inf, -4)U[2,8)
Solve algebraically 2 - (5/x-4) = (20/x^2 +4x)
1. Identify LCD --> (x)(x+4) 2. Multiply each term by LCD 2x(x+4) - x = 4 3. Get all terms on 1 side 2x^2 + 7x - 4 =0 4. Solve for the zeros (2x-1)(x+4) x = 1/2, x = -4 5. Check for extraneous solutions x = -4 is an extraneous solution x = 1/2 is a solution {1/2}
Solve the equation algebraically (7x / x+5) + (1/x-2) = (7 / x^2+3x-10)
1. Identify LCD --> (x+5)(x-2) 2. Multiply each term by LCD 7x(x-2) + x+5 = 7 3. Get all terms on 1 side 7x^2 - 13x - 2 =0 4. Solve for the zeros (7x+1)(x-2) x = -1/7, x = 2 5. Check for extraneous solutions x = 2 is an extraneous solution x = -1/7 is a solution {-1/7}
Solve the equation and check your answer q + 4 q-5 1 ------- + ------ = ----- 2 2 2
1. Identify LCD --> 2 2. Multiply each term by LCD q+4 + q-5 = 1 3. Get all terms on 1 side 2q-2 =0 4. Solve for the zeros q = 1 5. Check for extraneous solutions q = 1 is a solution {1}
Solve the equation and check your answer q + 10 q-11 1 ------- + ------ = ----- 6 6 6
1. Identify LCD --> 6 2. Multiply each term by LCD q+10 + q-11 = 1 3. Get all terms on 1 side 2q-2 =0 4. Solve for the zeros q = 1 5. Check for extraneous solutions q = 1 is a solution {1}
Solve the equation algebraically r+3 = 28/r
1. Identify LCD --> r 2. Multiply each term by LCD r^2 + 3r = 28 3. Get all terms on 1 side r^2 + 3r -28 =0 4. Solve for the zeros (r +7) (r -4) r = -7, r= 4 5. Check for extraneous solutions Both r=-7, and r=4 are a solutions {4,-7}
Solve (2/x-1) + x = 11
1. Identify LCD --> x-1 2. Multiply each term by LCD 2 + x(x-1) = 11(x-1) 3. Get all terms on 1 side x^2 - 12x + 13 =0 4. Solve for the zeros x = -(-12) +- sqrt((-12)^2 - (4 * 1 * 13) ---------------------------------- 2(1) x = 12 +- sqrt(92) ------------- 2 x= 6 +- sqr(23) 5. Check for extraneous solutions Both are solutions {6+sqrt(23), 6-sqrt(23)}
Solve ln(3x-2) + ln(x-1) = 2 ln x
1. Simplify ln (3x-2 * x-1) = ln x^2 2. Same base logarithms = each other (3x-2 * x-1) = x^2 3x^2 - 5x + 2 = x^2 2x^2 - 5x + 2 = 0 (2x - 1) (x -2) x = 2, x = 1/2 (extraneous) {2}
Solve the equation algebraically x + 9x/x-8 = 72/x-8
1. Identify LCD --> x-8 2. Multiply each term by LCD x^2 -8x + 9x = 72 3. Get all terms on 1 side x^2 + x -72 =0 4. Solve for the zeros (x+9)(x-8) x = -9, x = 8 5. Check for extraneous solutions x = 8 is an extraneous solution x = -9 is a solution {-9}
Solve: (2y/y+2) + (4/y) + 2 = (8/y^2 + 2y)
1. Identify LCD --> y(y+2) 2. Multiply each term by LCD 2y(y) + 4(y+2) = 8 3. Get all terms on 1 side 4y^2 + 8y =0 4. Solve for the zeros (4y) (y+2) y = 0, y = -2 5. Check for extraneous solutions Both y = 0, and y = -2 are extraneous solutions No Solution
Graph the function below f(x) = -ln(x-5)
1. Identify Transformations Right 5 units, reflect across the x-axis 2. Identify VA x-5 = 0 x = 5 3. Apply to Base points of logarithms (1,0) --> (6,0) (e, 1) --> (e+5 , -1) 4. Graph
Graph the polynomial function. f(x) = 44x^4 - 61x^3-6x^2 + 19x + 4
1. Identify basic shape based on degree 2. Identify end behavior 3. Find y intercept : 4 Graph
Find the partial fraction decomposition for the rational expression (70 / x^2-25)
1. Identify common denominator and form equation (70 / (x+5) (x-5) ) = ( A / (x+5) ) + ( B / (x-5) ) 2. Multiply each equation by common denominator 70 = A(x-5) + B(x+5) 3. Solve for B LET x = 5 70 = A(5-5) + B(5+5) 70 = 10B B = 7 4. Solve for A LET x = -5) 70 = A(-5-5) + B(-5+5) 70 = -10A A = -7 5. Plug into equation (70 / x^2-25) = ( -7 / x + 5) + (7 / x-5 )
Yawaka manufactures motorcycles and bicycles. To stay in business, the number of bicycles cannot exceed 5x the number of motorcycles. They lack the facilities to produce more than 100 motorcycles or more than 250 bicycles, and the total production of motorcycles and bicycles cannot exceed 300. If Yawaka makes $1280 on each motorcycle and $260 on each bicycle what should be made to maximize profits?
1. Identify constraints M >= 0 (Cannot have negative motorcycles) B >= 0 (Cannot have negative bikes) B <= 5M (Bikes cannot exceed 5x motorcycles) M <= 100 (Motorcycles must be less than 100) B <= 250 (Bikes must be less than 250) M + B <= 300 (Cannot exceed 300 together) P = 1280M + 260B 2. Graph and identify corners, and plug in for objective function Corners (M,B) ---> Value (0,0) --> 0 (100, 0) --> 128,000 (50, 250) --> 129,000 (100, 200) --> 180,000 3. Identify maximum 100 motorcycles and 200 bikes at $180,000
Find the remainder when f(x) = 8x^26 - 3x is divided by x+1
1. Identify k --> -1 2. Use the remainder therom to find the remainder by plugging k in for x 3. remainder is 11
Express as a difference of logarithms log 66/y
1. Identify rule log(R/S) = log R - log S 2. log 66 - log y
To solve equations in one variable
1. Identify the LCD 2. Multiply each term by LCD 3. Get all terms on 1 side 4. Solve for the zeros 5. Check for extraneous solutions by plugging zero in for the x in the LCD
Use the properties of logarithms to write an expanded expression log<5>y^2
1. Identify the rule log<b> R^c = c * log<b> R 2. 2 * log<5> y
Solving systems of inequalities procedure
1. Identify the type of border 2. Identify the type of line it is 3. Identify where to shade
Write parametric equations for the line through A and B A(3, 4, -5) B(-5, -4, 5)
1. Identify which is the point vector (the head of the line) A 2. Find vector AB (Second minus first) <-5 - 3, -4 -4, -5 - 5> AB <-8, -8, 10> 3. Write in vector equation format <x, y, z> = <3, 4, -5> + t<-8, -8, 10> 4. Convert to parametric x = 3 -8t y = 4 -8t z = -5 + 10t
Graph (calculator) the function in a viewing window that shows all of its extrema and x intercepts. Describe the end behavior using limits. zeros: (x-2), (x+3), (x+5)
1. Identify x int 2, -3, -5 2. Factor and plug into calculator 3. Draw picture with viewing window 4. Describe using end behavior lim x --> inf f(x) = inf lim x --> -inf f(x) = -inf
Plotting points with given polar coordinate
1. If negative magnitude, draw backwards 2. If negative angle start from 0 going backwards
Find the exact solution algebriacally log<6>(2x-7) = 4
1. If solving a logarithm with a argument = to something 6^4 = 2x-7 1296 = 2x-7 2x = 1303 x = 651.5 {651.5}
1.23^x = 5.7
1. If solving and exponent is a variable log 1.23^x = log 5.7 x * log 1.23 = log 5.7 x = log 5.7 / log 1.23 x = 8.408 {8.408}
A series will be convergent if ...
1. If the absolute value of the common ratio < 1
A series will be divergent if ...
1. It is an arithmetic series 2. If the absolute value of the common ratio is >= 1
When asked to write an infinitely repeating decimal as a fraction:
1. Seperate the repeating portion from the single occuring porition 2. Identify how many digits repeat and identify it as a power of 10 3. Plug numbers into formula: Single occurence + infEk=1 repeating digits * (1/power of ten) ^k 4. Get first three values 5. Solve first term / 1 - power of 10 + single occurence
Describe how to transform the graph of f(x) into the graph of g(x) Sketch the graph by hand and support answer with a grapher f(X) = e^x g(x) = e^-4x
1. List out transformations Horizonatly shrink by a factor of 1/4, Reflect over the y-axis 2. Apply transformations to the base points of an exponential function (0,1) and (1,b) (0,1) --> (0, 1) (1,b) --> (1,e) --> (-1/4 , -e) 3.Graph
Elimination Rules
1. Multiply 1 or both equations to cancel a variable 2. Solve for the other variable 3. Plug solved variable in for one of the original equations to solve for the canceled variable 4. Double check answer by plugging in both values for equation not used
Solve for x e^x - e^-x ------------ = 5 2
1. Multiply by 2 e^x - e^-x = 10 2. Multiply each term by [e]^x [e]^2x - 1 = 10[e]^x 3. Rearrange to a Quadratic [e]^2x - 10[e]^x - 1 4. Quadtratic formula [e]^x = Y Y^2 - 10Y - 1 Y= 5 +- sqrt(26) [e]^x = 5 + sqr(26) [e]^x = 5 - sqrt(26) ---> x can't be negative e^x = 5 + sqrt(26) ln 5 + sqrt(26) = x x = 2.312
In order to graph a plane in 3 dimensions ...
1. Set each term equal to the constant and solve (Find each intercept) 2. Plot intercepts 3. Draw solid lines between each point and extend them using dashed lines 4. Shade the triangle formed
When given two positions in a sequence without a1 and asked to find the explicit formula do ...
1. Set the first term = to b1 and solve for the common difference or common ratio 2. Use the common difference or common ratio to solve for a1 using the other term
Find the work done by a force of 30 N acting in the direction <2,2> in moving an object 3 m from (0,0) to a point on the line y = (1/2)x in the first quadrant
1. Not in the same direction, so |F| * |AB| * cos(x) 2. Find the angle between the two vectors by drawing a picture 1 line <2,2> and the other <2,1> 3. Use tan-1 to find angle of both triangles, and subtract tan-1(2/2) = 45 tan-1(1/2) = 26.6 final angle = 18.4 4. Plug into formula and solve 30 * 3 *cos(18.4) 85.399
Find the work done by a force of 12 N acting in the direction <1,2> in moving an object 4m from <0,0> to <4,0>
1. Not same direction, but a constant force in any direction = |F|*|AB| *cos(x) 2.Draw a picture based on the direction of the force 1rst quadrant over 1, up 2 3. Find angle of triangle using tan-1 tan-1 (2/1) = 63.4 degrees 4. Plug values into formula 12 * 4 * cos(63.4) 5. Solve 21.492
The angl between a 200lb force and AB = 2i +3j is 30 degrees. Find the work done by F moving an object from A to B.
1. Not same direction, so |F| * |AB| * cos(x) 2. Find magnitude of AB 2<1,0> + 3<0,1) <2,0> + <0,3> <2,3> sqrt( (2)^2 + (3)^2) = sqrt(13) 3. Solve 200 * sqrt(13) * cos 30 624.5
Determine whether the given ordered pair is a solution of the system of equations. (4,1) {9x -y = 37 {9x -5y = 31
1. Plug in values 9(4) - 1 = 37 9(4) - 5(1) = 31 2. Check (Must pass both equations to be a solution) 36 != 37 31 != 31
Find u*v given theta is the angle between u and v theta = 150 |u| = 3 |v| = 8
1. Plug numbers into formula cos 150 = (u*v) / 3 * 8 cos 150 = (u*v) /24 cos 150 * 24 = u*v u*v = -20.785
Rules to Solving Systems of equations using matricies
1. Put in AX = B form 2. Find the inverse of A A inverse = 1/(a*d)-(b*c) * [d -b] [-c a] 3. Place A inverse on the left of each side in the equation 4. Multiply A inverse by B if a 2x2 or plug into calculator if higher 5. Write answer
Describe the end behavior of the polynomial function using end behavior notation: f(x) = 6x^2 - x^3 + 4x - 2
1. Rearange -x^3 + 6x^2 + 4x -2 2. Describe using end behavior LC = -1 ; Negative = reflection, so opposite than normal Degree = 3; Odd degree = different directions lim x --> -inf f(x) = inf lim x --> inf f(x) = -inf
Solve for x where A = [3] [4] B = [9] [2] 3x + A = B
1. Rearrange to Solve for x, but remember no division of matrices 3x = B-A 1/3 (3x) = 1/3(B-A) x = 1/3(B-A) B-A = [6] [-2] Multiply by 1/3 x = [2] [-2/3]
Graph the function below f(x) = 4 + log(x)
1. Rewrite log(x) + 4 2. Identify transformations Up 4 units 3. Identify VA x = 0 3. Apply transformation to base points (1, 0) --> (1, 4) (10, 1) --> (10, 8) 4. Graph
Graph the function below f(x) = log(-3 - x)
1. Rewrite -(3 + x) --> -(x + 3) --> log(-x - 3) 2.Identify transformations Right 3, Reflect across y-axis 2.5 Identify VA -x-3 = 0 -x = 3 x = -3 3. Apply to base points of logartihms (1,0) and (b,1) (1,0) --> (-4,1) (10,1) --> (-13, 1) 4. Graph
Use completing the square to describe the graph of each function: f(x) = 10 - 16x - x^2
1. Rewrite in standard form -x^2 - 16x + 10 2. Move constants to the right y - 10 = -x^2 -16x 3. Divide by -1 y-10/-1 = x^2 + 16x 4. Split x^2 y-10/-1 = (x + 8)^2 + 64 y-10/-1 - 64 = (x +8)^2 5. Multiply by -1 y-10 + 64 = -(x+8)^2 6. Add constants to right y = -(x+8)^2 + 74
Find the work done by lifting a 2600lb car 5.5ft
1. Same direction = |F| * |AB| 2600 * 5.5ft W = 14300
Find the zeros of the function algebraically: f(x) = 7x^3 - 26x^2 - 8x
1. See if you can factor out anything x (7x^2 - 26x - 8) 2. Factor x (7x + 2) (x -4) 3. Set factors equal to 0 x = 0 7x + 2 = 0 x = -2/7 x-4 = 0 x = 4
Find the zeros of the function algebraically: f(x) = x^2 + x - 30
1. See if you can factor out anything - no 1.5 Factor (x + 6) (x - 5 ) 2. Set factors equal to zeros x + 6 = 0 x = -6 x - 5 = 0 x = 5
Find the zeros of the function algebraically: f(x) = 6x^2 + 25x - 9
1. See if you can factor out anything - no 2. Factor (3x - 1) (2x + 9) 3. Set factors equal to zero 3x - 1 = 0 x = 1/3 2x+9 = 0 x = -9/2
Use an algebreic method to eliminate the parameter x = 2t - 3, y = 9 -4t; 3<= t <= 5
1. Solve 1 for t x = 2t-3 x + 3 = 2t/2 x/2 + 3/2 = t 2. Substitute y = 9 - 4(x/2 + 3/2) y = 9 -2x - 6 3. Write answer replacing restriction t with solved t y = -2x + 3; <3<= x/2 + 3/2 <= 5
Solve: {y = x^2 -4x -1 {4y - x = 1
1. Solve for a variable 4y - x = 1 4y = x + 1 y = 1/4x + 1/4 2. Substitute and Solve for the other variable 1/4x + 1/4 = x^2 - 4x -1 4(1/4x + 1/4) = 4(x^2 -4x -1) x + 1 = 4x^2 -16x -4 4x^2 - 17x -5 x = -(-17) +- sqrt( (-17)^2 - 4(4)(-5) --------------------------------- 2(4) x = 4.53 , x = -0.28 3. Plug into original equation y = (4.53)^2 -4(4.53) -1 y = (-0.28)^2 -4(-0.28) -1 y = 1.40 y = 0.2 4. Write as an ordered pair {(-0.28, 0.2), (4.53, 1.4)}
Use an algebreic method to eliminate the parameter x= t^2, y = t + 1
1. Solve for t sqrt(x) = t 2. Substitute y = sqrt(x) +1
x = 2sin t, y = 2 cos t ; 0<= t <= 3pie/2
1. Solve t sin -1 (x/2) = t y = 2 cos ( sin-1(x/2) ); 0 <= sin-1(x/2) <= 3pie/2
State the degree and list the zeros of the polynomial function. State the multiplicity of each zero, and whether the graph crosses the x-axis at the location Sketch a graph f(x) = x (x-3)^2
1. State the degree by adding the multiplicities of each factor 2 + 1 = 3 degree = 3 2. State the multiplicity of each zero and whether it crosses the x-axis Multilicity of x = 1; crosses x axis because it is odd Multiplicity of (x-3)^2 = 2; touches x axis because even 3. Reference sheet for graph
Use an algebreic method to eliminate the parameter x = t, y = t^3 -2t +3
1. Substitute y = (x)^3 -2x +3
Use an algebreic method to eliminate the parameter x = 1 + t, y = t
1. Substitution x= 1 + y 2. Write in slope intercept form y = x -1
Completeing the square
1. Subtract constants from right to left If division divide both sides by a 2. Split x^2 +- constant into (x +- constant /2)^2 3. Add constant /2 squared to the left If division multiply both sides by a 4. Move constant to the right
Use completing the square to describe the graph of each function: f(x) = x^2 - 4x + 6
1. Subtract constants from right to left y - 6 = x^2 - 4x 2. Split x^2 y - 6 = (x - 2)^2 +4 3. Add constant squared to the left y - 2 = (x - 2)^2 4. Move constants to the right y = (x-2)^2 + 2
How to find end behavior asymptotes
1. Sythetically divide the numerator by the denominator and find the remainder 2. Drop the remainder and the rest of the division is the end behavior asymptotes
A person is sitting on a sled on the side of a hill inclined at 60 degrees the person and the sled weigh 160lbs. What is the force required to keep the sled from sliding down the hill.
1. Think of problem as a gravitational force holding the sled. 2. Find the gravitational force using mag * sin(x) 160 * sin(60) F = 138.564
In a three dimensional graph how is the z line drawn top to bottom
Solid --> Vertex --> Dashed
If D = 11300 - 60p and S = 300 + 50p are demand and supply equations respectively, find the equilibrium point
1. To find an equilibrium point set the equation = to each other and solve for variable 11300 - 60p = 300 +50p 11000 - 60p = 50p 11000 = 110p p = 100 2. Plug variable into either equation to find y value D = 11300 -60(100) D = 5300 3. Write as an ordered pair {(100, 5300)}
When given an equation in y = mx+b format and asked for parametric form then...
1. Turn y into <x,y>, x into t, and b into 1 of the two starting points 2. Turn m into a slope vector by finding the distance between the x values and the distance between the y values <x1 --> x2, y1 --> y2> <sx, sy> 3. <x,y> = <sx,sy>t + (px, py) 4. Write as a parametric {x = sx*t + px} {y = sy*t + py}
Write the expression using only natural logarithms log<5> x
1. Use change of base ln x ---- ln 5
Write the expression using only common logs log<5> x
1. Use change of base log x ----- log 5
Describe how to transform the graph of ln x into the function given below f(x) = log<1/4> x
1. Use change of base to see how ln x turned into log<1/4> x ln (x) / ln (1/4) ln(x) / ln 1 - ln 1/4 --> ln(x) / -ln(4) -1/ln(4) * (ln x) 2. Identify Transformations Vertical shrink by a factor of 1/ln(4), and reflected across the x-axis 3. Identify VA: x = 0 4. Apply Transformations (1,0) --> (1 , 0) (e, 1) --> (e , -0.721) 5. Graph
Find the dot product of u and v u = <4,5> v = <-3, -7>
1. Use dot product formula to find dot product 4 * -3 + 5 * -7 u*v = -47
Find the dot product of u and v u = <5,3> v = <12,4>
1. Use dot product formula to find dot product 5* 12 + 3*4 u*v = 72
Find the rectangular coordinate of (3, 2pi/3)
1. Use formula x = r*cos(theta) and y = r*sin(theta) x = 3 * cos(2pi/3), y = 3* sin(2pi/3) x = 3 *-1/2, y = 3*sqrt(3) /2 x = -3/2, y= 3*sqrt(3) / 2
Find the rectangular coordinate of (-2, 60)
1. Use polar to rectangular formula x= r*cos(theta) y = r*sin(theta) x = -2 * cos 60 ; y = -2 * sin 60 x = -2 * 1/2 ; y = -2 * sqrt(3)/2 x = -1 ; y = -sqrt(3)
Given 2 points, write a function in vertex form: Vertex: (-1, -3) and (1,5)
1. Using Vertex you know h and k, so plug into form y = a(x+1)^2 -3 2. Plug in point; Simplfy and solve for a 5 = a(1+1)^2 - 3 5 = a(2)^2 - 3 5 = 4a - 3 a = 2 3. Finish pluging into formula y= 2(x+1)^2 - 3
Let vector U = <-1, 3> and vector W = <2, -5> Find the compontent form of the vector U - W
1. Using vector arithmitic simply subtract the first terms and the last terms <-1, 3> - <2, -5> --> Vector U-W = <-3, 8> Vector U+V = <-3, 8>
To go from an vector equation to parametric equations ...
1. Variable = variable of the point vector + variable of the directional vector * t <x,y,z> = <1, 4, 5> + t<4, 7, 9> Ex: x = 1 + 4t 2. Repeat for all variables
How to Find Undefined Values
1. What makes the number underneath the radical negative If it makes a radical negative you use an in equality such as x < # to describe it, then proceed to cross out all the numbers behind it. 2. What makes the denominator zero
Covert polar equation to rectangular equation and identify the graph r cscx = 1
1. When cscx or secx is on the left then -divide by csc or sec -Multiply by r -Simplify r = 1/csc x r = sin x r^2 = rsinx x^2 + y^2 = y X^2 + Y^2 + y = 0 x^2 + (y - 1/2)^2 = 1/4
Find vector v given u = <2,3>, u*v = 10, |v|^2 = 17
1. Write an equation for the dot product, Solve for one of the two variables 2x + 3y = 10 2x = 10 - 3y x = - 3/2y + 5 2. Write the magnitude of v as an equation |v| ^2 = sqrt( x^2 + y^2) ^2 = 17 x^2 + y^2 =17 3.Plug in solved variable (-3/2y + 5)^2 + y^2 = 17 4. Simplify (-3/2y + 5) * (-3/2y + 5) + y^2 9/4y^2 -15/2y -15/2y + 25 + y^2 = 17 13/4y^2 -15y + 8 = 0 4. Remove denominators by multiply all by 4 13y^2 -60y +32 5. Plug into quadratic formula x = -(-60) +- sqrt( (-60)^2 - 4(13)(32) / 2(13) y = 4 or 8/13 6. Plug y into 1 of the equations to find the x value and list as vector v 2x + 3(4) = 10 2x + 12 =10 2x = -2 x= -1 v = <-1, 4> or 2x + 3(8/13) = 10 2x +24/13 = 10 2x = 106/13 x = 53/13 v = <53/13 , 8/13>
Evaluate the logarithimic expression: log 1/10
1. Write as exponential 1/10 = 10^x 10^-1 = 1/10 log 1/10 = -1
Let vector U = <-1, 3> and vector V = <2,4> Find the component form of the vector U+V
1.Using vector arithmitic simply add the first terms together and the last terms together <-1, 3> + <2,4> --> vector U+V = <1, 7> vector U+V = <1, 7>
Expand (z^-2 + z^2)^4
1/x^8 + 4/x^4 + 6 + 4z^4 + z^8
In a three dimensional graph the how is the x line drawn bottom through vertex to top
Solid --> Vertex --> Dashed
When given restrictions for t for graphing
Change t max and t min
Guassian Elimination
Choose 1 equation and eliminate the variable in that equation from every other equation Repeat until you have one variable = to a constant, and then solve using substitiution Make sure to check with the original equations
How many 4-person commitees can be formed out of 20 students
Combination There are 4845 possible 4-person committees
An airplane is flying on a bearing of 335 at 530 mph. Find the component form of the velocity of the airplane
Convert bearing angle to a trig angle: Draw the 335 degree angle in bearings Find the angle in triangle - 65 degrees Identify the quadrant - (2nd Qaudrant) Apply appropriate formula - 180 - a Trig angle is 115 Find the component form: <mag*cos(trig angle) , mag*sin(trig angle)> <530 * cos(115) , 530 * sin(115)> <-223.988, 480.343>
Find the domain of the function f(x). Use limits to describe the end behavior at values of x not in the domain. f(x) = -1/x^2 -16
D:(-inf, -4) U (-4,4) U (4, inf) lim x --> -4 - f(x) = -inf lim x --> -4 + f(x) = inf lim x --> 4 - f(x) = inf lim x --> 4+ f(x) = -inf
Draw the inequality with its graph, indicate whether the boundary is included in or excluded from the graph y > 3
Dotted horizontal line Shaded above Reference System of inequalities #1
Distance from focus to focus in a hyperbola
Focal Axis
Describe how the graph of f(x) = 2/x-3 can be obtained by transforming the graph of the recipricol function g(x) = 1/x. Identify the horizontal and vertical asymototes and use limits to describe the cooresponding behavior. Sketch the graph of the function
From 1/x right 3 units, then vertically stretch by a factor of 2. VA: x = 3 HA: y = 0 lim x --> 3 - f(x) = -inf lim x --> 3 + f(x) = inf lim x --> inf f(x) = 0 lim x --> -inf f(x) = 0 Refer to graphs of rational functions for sketch #5
Describe how the graph of f(x) = 2/x-6 can be obtained by transforming the graph of the recipricol function g(x) = 1/x. Identify the horizontal and vertical asymototes and use limits to describe the cooresponding behavior. Sketch the graph of the function
From 1/x right 6 units followed by a vertical stretch by a factor of 2. VA: x = 6 HA: y = 0 lim x --> 6 - f(x) = -inf lim x --> 6+ f(x) = inf lim x --> inf f(x) = 0 lim x --> -inf f(x) = 0 Refer to graphs of rational functions for sketch #6
Row Echelon Form Rules
Get a diagnol of 1s with 0s underneath them
Reduced Row Echelon Form
Get a diagnol of 1s with 0s above and below them
Going from rectangular coordinates to polar coordinates
Given (x,y) find (r, x) r = sqrt( (x)^2 + (y)^2 ) x = tan-1 (y / x) x can be multiple things in polar coordinates, so draw a picture with the angle found. restriction -2pie < x <2pie
Horizontal asymptote format
HA: y = #
Use the discriminant to determine whether the graph of the function is an ellipse/circel, a hyperbola, or a parabola: x^2 - 8xy - 2y^2 - 7 = 0
Hyperbola
Write the statement as a power function equation The current I in an electrical circuit is inversly proportional to the resistance R, with a constant of variation V
I = V/R
If you multiply a matrix by its inverse you get the
Identity Matrix
Discriminant Test
If B^2 - 4AC > 0 --> hyperbola If B^2 - 4AC = 0 --> parabola If B^2 - 4AC < 0 --> ellipse
The remainder therom
If f(x) / x - k then f(k) will give you the remainder
Convert the rectangular equation to polar form 2x - 3y = 5
If given both x and y, and not a circle then - convert x and y to r format - factor out r - divide by what is left in parentheses 2x - 3y = 5 2rcos x - 3rsin x = 5 r(2cosx - 3sinx) = 5 r = 5/(2cosx - 3sinx)
Convert the rectangular equation to polar form (x-3)^2 + (y+3)^2 = 18
If given in circular form then -Factor -Simplify and subtract constant to get right side = 0 -Combined x^2 and y^2 to make r^2 -Make other variables their respective r form -Divide by r -Get r alone (x-3)(x-3) + (y+3)(y+3) = 18 x^2 - 6x + 9 + y^2 +6y + 9 = 18 x^2 + y^2 - 6x + 6y = 0 r^2 - 6rcosx + 6rsinx = 0 r - 6cosx + 6sinx = 0 r = 6cosx - 6sinx
Convert the rectangular equation to polar form (x-3)^2 + y^2 = 9
If given in circular form then -Factor -Simplify and subtract constant to get right side = 0 -Combined x^2 and y^2 to make r^2 -Make other variables their respective r form -Divide by r -Get r alone (x-3)(x-3) + y^2 = 9 x^2 -6x + 9 + y^2 = 9 x^2 + y^2 -6x = 0 r^2 -6rcosx = 0 r - 6cosx = 0 r = 6cosx
Convert the rectangular equation to polar form x = 2
If given x or y change to its respective r format rcosx = x rsinx = y x = 2 rcosx = 2 r = 2/cosx r = 2 * 1/cosx r = 2 sec x
Determine whether the sequence converges or diverges. If it converges give the limit 4n-4 / 5 - 2n
If numerator degree > denominator --> Diverges If numerator degree = denominator --> top / bottom coefficent If numerator degree < denominator --> Limit is 0 4/-2 --> -2 lim n --> inf = -2
Use synthetic division to check that the number k is an lower bound for the real zeros of the function: f(x) = 5x^3 - 7x^2 + x -5 k = -7
It is a lower bound Alternating non-positive numbers
Determine if the function is a monomial function, if it is then state the degreee and the leading coefficent. If not explain why. f(x) = -6x^7
Monomial Function Degree = 7 Leading Co = -6
Determine if the function is a monomial function, if it is then state the degreee and the leading coefficent. If not explain why. f(x) = -4
Monomial Function Degree = O Leading Co: -4
Difference between monomial function and power function
Monomial Functions -Can only have positive integer exponents -Exponents called degree -Monomials can be constants, power's can't Power Functions -Can have any number as an exponent except 0 -Exponents are called powers
Does order matter in combinations
No
If all zeros are extraneous solutions the function is said to have ...
No Solution
Determine if the function is a monomial function, if it is then state the degreee and the leading coefficent. If not explain why. f(x) = 3x^-5
Not a Monomial Function No negative exponents
Determine if the following is a polynomial function. If it is then state the degree and leading coefficent; if it is not then explain why not. f(x) = 3rt(27x^3 + 8x^6)
Not a Polynomial Polynomials do not have roots
Determine if the function is a power function, if it is then state the power and the constant of variation. If not explain why. f(x) = 3 * 2^x
Not a Power Function Must follow y = kx^a - x cannot be an exponent
Area with permiter problems
Permiter = 2w + 2L Area = W*L so write as 2w + 2L = P w * L = A then set = to values and solve if needed
Systems of equations with different slopes ...
Only intersect once
When does vector v = vector w v = <v1, v2, v3>, w = <w1, w2, w3>
Only when v1 = w1 v2 = w2 v3 = w3
Sketch a graph of the equation. Label all intercepts. x + 2y + 3z = 8
Orange Notebook; 3D Cartesian Coordinate System #8
Draw the inequality with its graph, indicate whether the boundary is included in or excluded from the graph 2x - 5y >= 2
Solid Line Shaded below Reference System of inequalities #2
Use the discriminant to determine whether the graph of the function is an ellipse/circel, a hyperbola, or a parabola: 4x^2 - 12xy + 9y^2 - 6x + 5y = 0
Parabola
the set of points that are equidistant from a line and a point
Parabola
If the solved variable in a substitution/elimination proble has 2 answers then
Plug in both answers into seperate equations and solve for the other variable list as {(x1, y1), (x2,y2)}
Determine if the following is a polynomial function. If it is then state the degree and leading coefficent; if it is not then explain why not. f(x) = 2x^5 - 1/2x + 9
Polynomial Degree: 5 Leading Coefficent: 2
Determine if the function is a power function, if it is then state the power and the constant of variation. If not explain why. f(x) = 1/2x^5
Power Function Power = 5 Constant = 1/2
Determine if the function is a power function, if it is then state the power and the constant of variation. If not explain why. f(x) = 9x^5/3
Power Function Power = 5/3 Constant = 9
Describe how to transform the graph x^2 to the given function. f(x) = (x-3)^2 - 2
Right 3 Down 2
Summation notation can also be written as ...
Sn where n is the number of terms that are added
Plot the point whose coordinates are (-3, 0, -5)
Refer to Orange Notebook 3D Cartesian System #1
Graph: x^2/25 - y^2/9 = 1
Refer to Orange Notebook Conic Sections Hyperbolas #4
Graph: (y-1)^2/25 - (x+1)^2/4 = 1
Refer to Orange Notebook Conic Sections Hyperbolas #5
Graph: (x-3)^2/4 - (y+2)^2/9 = 1
Refer to Orange Notebook Conic Sections Hyperbolas #8
Find the minimum and maximum if they exist of the objective function f, subject to the constraints Objective function: f = 4x + 3y Constraints: x + y <= 9 x - 2y <= 0 x >= 0 y >= 0
Reference Linear Programming #7.2 Minimum of 0 at (0,0) Maximum of 33 at (6,3)
Use the remainder therom to find the remainder when f(x) is divided by k. f(x) = x^3 - 3x^2 + 4 k = -2
Remainder = -16
Rewrite the equation in vertex form: f(x) = 8x - x^2 + 3
Rewrite in Standard form: -x^2 + 8x + 3 1. Find h using -b/2a h = -8 / 2(-1) h = 4 2. Find k by plugging h into standard form f(4) = -(4)^2 + 8(4) + 3 f(4) = 19 3. Identify a as leading coefficent a = -1 3. Plug into Vertex form y = -(x - 4)^2 + 19
a * cosbx or a * sinbx
Rose Curve If b is odd, then it is the number of pedals If b is even, then the number of pedals is 2b a makes bigger or smaller
Identify the graph r = 2sin3x
Rose curve with 3 pedals
If the series is convergent the formula for the sum of an infinite geometric series is:
S inf = a1/1 - r
Find the sum Sn of the first 80 terms of the arithmetic sequence: 1, 3, 5, 7, ...
S80 = 6400
the sums of all terms in a sequence
Series
Where do you shade for an inequality
Shade where the test point is true
Draw the inequality with its graph, indicate whether the boundary is included in or excluded from the graph x^2 + y^2 <= 49
Solid circle with a radius of 7 and a center of (0,0) Shaded inside Reference System of Inequalities #7
Identify the graph r = 2x
Spiral
Baby Lucy wants to arrange 7 blocks in a row. How many different arrangements can the baby make?
The baby can make 5040 different arrangements of the block
When given an fraction exponent ...
The numerator speccifies the Shape and even or odd The denominator specifies the root If the root is even then the left hand side does not exist, If the root is odd then the left hand side does exist
Given n = c/v; Write a description of the power function n = the refractive index of a medium c = the constant velocity of light in a free space v = the velocity of light in a medium
The refractive index of a medium N is inveresly proportional to the velocity of light in a medium V, with the constant velocity of light in a free space C as the constant of variation.
Given f(x) = x^.73
The shape of the graph will be a square root function
Given w = mg; Write a description of the power function W = weight M = mass G = the constant acceleration due to gravity
The weight W is directly proportional to the mass M, with the constant of variation being G, the constant acceleration due to gravity
How many distinguishable 6 letter words can be formed using the letters in "HAWAII"
There are 180 distinguisable 6 letter words that can be formed using the letters in "HAWAII" 6! / 1! * 2! * 1! * 2! number of letters! / each letter! * each other letter!
How many different 7 card hands include the ace, jack, and king of spades.
There are 211,876 different 7-card hands that include these cards by default
An identification code is to consist of 2 letters followed by 4 digits. How many different codes are possible if repetition is not permitted
There are 3276000 possible identification codes following this pattern
Distance from vertex to vertex in a hyperbola
Transverse Axis
Use Gaussian Elimination to solve the linear system by finding an equivilant system in triangular form. {x + y -z = 1 {y + w = -4 {x - y = 3 {x + z + w = 4
Triangular form {x + y -z = 1 {y + w = -4 {2w + z = -6 {w = -11/2 Solution {(19/2, 3/2, 5, -11/2)}
If log<b>U = log<b>V then
U = V
Evaluate the logarithimic expression: ln (e^-5)
Use ln e^n = n ln (e^-5) = -5
Change of Base -Uses and formula
Used to change logs with a base to common or natural logs log<b> N = (log<c> N) / (log<c> b)
Evaluate the logarithimic expression: log<11> 11
Using log<b> b = 1 log<11> 11 = 1
Write the statement as a power function equation The volume V of a circular cylinder with fixed height is proportional to the square of its radius r
V = k * r^2
Find the verticies and foci of the ellipse x^2/16 + y^2/12 = 1
V: (-4,0) , (4,0) F:(-2,0), (2,0)
Prove that the graph of the equation is an ellipse and find its verticies, foci, and eccentricity 16x^2 + 64y^2 -96x+384y - 304
V: (11, -3) , (-5, -3) F: (3 + 4sqrt(3), -3) , (3 - 4sqrt(3), -3) e = sqrt(3)/2
Find the vertex, focus, directrix, and focal width of the parabola: x^2 = 4y
Vertex: (0,0) Focus:(0,1) Directrix: y = -1 Focal Width = 4
Find the Vertex and the axis of the function: f(x) = 5(x-1)^2 - 7
Vertex: (1, -7) Axis: x = 1
Find the vertex and the axis of the function: f(x) = 3(x-1)^2 + 5
Vertex: (1, 5) Axis: x= 1
If x^2 is on the left of a parabola it means
Vertical
a under y^2 in a hyperbola means
Vertical
a under y^2 in an ellipse means
Vertical
Describe how to obtain the graph of the fiven monomial functional from the graph x^n where n is the same as the degree. State whether the function is even or odd. f(x) = 2/3x^4
Vertical shrink of magnitude 2/3 Even function
Describe how to obtain the graph of the fiven monomial functional from the graph x^n where n is the same as the degree. State whether the function is even or odd. f(x) = -1.5x^5
Vertical stretch of magnitude 1.5, Reflection accross the x axis Odd function
If force is a constant magnitude in any direction then...
W = F*AB (dot product) or W = |F| * |AB| * cos(x)
An airplane took 3 hours to fly 1800 miles against a head wind. The return trip with the wind took 2 hrs. Find the speed of the plane in still air and the speed against the wind.
W = Speed of Wind S = Speed of Plane 1. Find the rate for both situations up: r = 1800/3 --> 600 down: r = 1800/2 --> 900 2. Write equations and solve for 1 variable Down: S + W = 900 S - W = 600 Solve for S by getting W by itself W = 900 - S W = -600 + S 900 - S = - 600 + S 2S - 600 = 900 2S = 1500 S = 750 Solve for W by getting S by itself S = 900 - W S = 600 + W 900 - W = 600 + W 2W + 600 = 900 2W = 300 W = 150 Speed of Plane: 750 mph Speed of Wind = 150 mph
Write the function as a product of linear and irredusible quadratic factors with all real coefficents f(x) = x^3 -14x^2 -8x -105
Zero: 15 Factor form: (x - 15)(x^2 + x + 7)
Determine the x value that causes the polynomial to a) be zero b) be positive c) be negative f(x) = (x+4)(x+2)(x-1)
a) Zeros: x = -4, x = -2, x = 1 b) f(x) > 0: (-4, -2)U(1, inf) c) f(x) < 0: (-inf, -4) U (-2, 1) 0 0 0 <------|---------|-----------------|--------------------> -4 -2 1 Test conditions x = -5 (-)(-)(-) == - x = -3 (+)(-)(-) == + x = 0 (+)(+)(-) == - x = 2 (+)(+)(+) == +
Convex limacon
limacon: a/b >= 2
Evaluate the logarithimic expressionln ln 9throot(e^7)
ln (e^7)^1/9 ln (e^7/9) ln 9throot(e^7) = 7/9
Use the properties of logarithms to expand the logarithmic expression as much as possible log cbrt(x/z)
log (x/z) ^ 1/3 log (x^1/3 / z^1/3) log x^1/3 - log z^1/3 1/3 log x - 1/3 log z
Assuming x, y, z are positive use the properties of logarithms to write the expression as a single logarithm 8 log (xy) - 7 log(yz)
log(xy)^8 - log(yz)^7 log(x^8 y^8) - log(y^7 z^7) log(x^8 y^8) / (y^7 z^7) log(x^8 y) / z^7
Properties of all logarithmic functions
log<b> (R*S) = log<b>R + log<b>S log<b>(R/S) = log<b>R - log<b>S log<b> R^c = c * log<b> R
Properties of Log, Natural Logs, Common Logs
log<b> 1 = 0 ; ln 1 = 0; log 1 = 0 log<b> b = 1; ln e = 1; log 10 = 1 log<b> b^n = n; ln e^n = n; log 10^n = n b^log<b> m = m; e^ ln m = m; 10 ^ log m = m
Rational zeros Therom
narrows down the possible rational zeros p/q p is the constant q is the leading coefficent
Divide f(x) by d(x) and write a summary statement in polynomial form and fraction form f(x) = x^2 -14x + 57 d(x) = x-7
polynomial form: x^2 -14x + 57 = (x-7)(x-7) + 8
polynomial form: Ex: 2x^4-x^3 - 2 ------------- 2x^2 + x + 1
top = divisor * quotient + remainder Ex: 2x^4-x^3 - 2 ------------- 2x^2 + x + 1 2x^4 - x^3 -2 = (2x^2 + x +1) * (x^2 - x) + (x -2)
In a hyperbola the graph never ...
touches the asymptotes
Triangular form system of equations produce a
triple pair
If b^u = b^v then
u = v
Unit Vector equation (v) v = <v1, v2, v3>, w = <w1, w2, w3>
u = v/|v| u = <v1/|v| , v2/|v| , v3/|v|>
The degree of a Zero function is
undefined
Addition of vectors v = <v1, v2, v3>, w = <w1, w2, w3>
v + w = <v1+ w1, v2+ w2, v3 + w3>
f(x) <= 0
when the function has y values less than or equal to 0
Traingular form is
where every variable has one equation
When it is a horizontal parabola, p affects the _ value
x
The dot product of two vectors is
x * x + y * y
Graph the function f(x) = 11x^2 + x -11 / x^2 -1 Find all asymptotes. List x and y intercepts
x-int: x = 0.96, x = -1.05 y-int: y = 11 VA: x = 1, x = -1 lim x --> 1 - f(x) = -inf lim x --> 1 + f(x) = inf lim x --> -1 - f(x) = -inf lim x --> -1 + f(x) = inf HA: y = 11 lim x --> -inf f(x) = 11 lim x --> inf f(x) = 11 Refer to graph on Asymptotes and Intercepts # 5
Find an equation in standard form for the ellipse that satisfies the given conditions: Major axis length on y axis = 14 Minor axis length = 2 Center (0,0)
x^2/1 + y^2/49 = 1
Find an equation in standard form for the ellipse that satisfies the given conditions: Endpoints of axes are (+-7,0) and (0,+-5)
x^2/49 + y^2/25 = 1