16 Week Pre Calc Exam

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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)


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