16 Week Pre Calc Exam
The Transverse Axis =
2a
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)
To prove an ellipse equation:
1. Complete the square by seperating x and y squared terms
Solving systems of inequalities procedure
1. Identify the type of border 2. Identify the type of line it is 3. Identify where to shade
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
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
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
In a three dimensional graph how is the y line drawn left to right
Dashed --> Vertex ---> Solid
Tvertical parabola centered at (h,k)
Equation: (x-h)^2 = 4p(y-k) Directrix: y = # (p in the opposite direction from the center)
Reduced Row Echelon Form
Get a diagnol of 1s with 0s above and below them
Plot the point whose coordinates are (-3, 0, -5)
Refer to Orange Notebook 3D Cartesian System #1
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
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
Partial Fraction Decomposition Format
a+b/c*d = ?/c + ?/d
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
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)
The order of a matrix
m x n number of rows x number of columns
In a three dimensional graph dashed lines are
negative values
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
Substitution using triangular form
plug the single variable into the second equation and then both into the first
Calculate the expression using r = <3, 0, -1> and v = <-7, 5, -6> r + v
r+v = <-4, 5, -7>
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
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
equation for a horizonal hyperbola centered at (0,0)
x^2/a^2 - y^2/b^2 = 1
When it is a vertical parabola, p affects the _ value
y
Find the partial fraction decomposition for the rational expression (20 / x^2-4x)
(20 / x^2-4x) = ( -5 / x) + (5 / x-4)
Foci in an ellipse
(c, 0) and (-c,0) relative to the center
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 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 (-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
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
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
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]
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
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]
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}
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
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)}
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)}
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
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
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
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
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
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
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
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
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
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]
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
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
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
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)}
The point vector is
1. The only point given OR 2. The head of the directional vector
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)}
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
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)
To graph an ellipse equation you need
2 vertices 2 points on the minor axis Box connecting points and draw curvy lines
Difference of focal radii of a Hyperbola
2a
Major axis =
2a
Sum of Focal Radii =
2a
Minor axis =
2b
The Conjugate axis =
2b
The Focal Axis =
2c
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)}
Evaluate r*(v+w) r = <-1, 4, 1> v = <2, 2, 4> w = <-4, 7, 4>
46
How many points are needed to graph a parabola?
5 1 vertex 2. Focal endpoints 2 random points found by plugging into equation
Evaluate w/|w| w = <12m -3, -4>
<12/13, -3/13, -4/13>
Vector equation:
<x,y,z> = <point vector> + t(directional vector>
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]
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
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) )
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)
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
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
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)
Distance from focus to focus in a hyperbola
Focal Axis
Row Echelon Form Rules
Get a diagnol of 1s with 0s underneath them
Systems of equations with the same equation...
Have infinitely many solutions
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
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
the set of all points whose distance's from two fixed points (Foci) have a common difference
Hyperbola
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
If result of substitution/elimination is 0=0 the solution is ...
Infinite Solutions It is written as {(x,y) | one of the equations}
How many solutions does a system of equations with parallel lines have?
No Solutions
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
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
Graph the system of inequalities {x^2 + y^2 <= 100 { y >= x^2
Parabala and circle Reference Linear Programming #3
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
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
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)}
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
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
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)
Where do you shade for an inequality
Shade where the test point is true
In a three dimensional graph how is the z line drawn top to bottom
Solid --> Vertex --> Dashed
In a three dimensional graph the how is the x line drawn bottom through vertex to top
Solid --> Vertex --> Dashed
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
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
Elementary Row Operations
Swapping, Adding to Replace, and Dividing by an common factor
If the equation is in the form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 you use what?
The Discriminant Test
In a hyperbola a^2 is always what
The first denominator
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
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)}
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 verticies and foci of the ellipse 8x^2 + 9y^2 = 72
V:(3,0), (-3,0) F: (-1,0),(1,0)
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
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)
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
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]
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
equation to find c in an ellipse
a^2 = b^2 + c^2
Identity matrixs are always
diagonal 1s filled in with 0s on the side
> or < get ____________ borders
dotted
For ellipses a must be
greater than b
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>
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)
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
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
>= or <= get __________ borders
solid
If result of substitution/elimination is 2 constants that don't equal each other then
there is no solution
In a hyperbola the graph never ...
touches the asymptotes
Triangular form system of equations produce a
triple pair
Unit Vector equation (v) v = <v1, v2, v3>, w = <w1, w2, w3>
u = v/|v| u = <v1/|v| , v2/|v| , v3/|v|>
Addition of vectors v = <v1, v2, v3>, w = <w1, w2, w3>
v + w = <v1+ w1, v2+ w2, v3 + w3>
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
Traingular form is
where every variable has one equation
When it is a horizontal parabola, p affects the _ value
x
In a three dimensional graph the x,y,z coordinates go
x --> Diaganolly y --> horizontally z --> Vertically
Find the equation of the parabola with a focus at (0,12) and directrix y = -12
x^2 = 48y
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
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
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
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
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 linear systems of equations by computing X = A^-1B {x + 4y = -24 {6x + 5y = -49
{(-4, -5)}
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)}
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)}
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
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)