Adv Alg / Trig Name S1 Final Exam Review

Unit 1: Linear Functions & Regression I can: Determine the average rate of change I can: Graph an equation or inequality

( ) ( ) ( )f b f a

b af x

−

−=

Slope Intercept Form: y mx b= +

1) Plot the y – intercept 2) Move according to the slope

Standard Form: Ax By C+ =

1) Find the x – intercept Ax C= → ( )0x,

2) Find the y – intercept By C= → ( )0,y

3) Plot intercepts and connect!

Point Slope Form: ( )1 1y y m x x− = −

1) Form a point with → ( )1 1x ,y

2) Find that point 3) Move away from that point according to the slop

#1 – 2: Find the average rate of change each of the following problems.

1. Given 2 8f ( x ) x= − , show all work to determine the average rate of change between 1− and 5 .

Work:

Average rate of change:

2. Given 1

52

f ( x ) x= − + , show all work to determine the average rate of change between 4 and 9.

Work:

Average rate of change: #3 – 8: Graph each equation or inequality.

3. 1

25

y x= − + 4. 2 6y x= − 5. 2

5 13

y ( x )+ = −

6. 2 5x y− ≤ 7. 3 5 15x y− > 8. 3y =

#9 – 11: Write an equation of the line. 9. 10. 11.

I can: Write an equation of a line when given a...

I can: Write an equation of a line given …

Slope and a Point:

1) Substitute in: ( )1 1y y m x x− = −

Two Points:

1) Calculate the slope: 2 1

2 1

y yx xm

−−=

2) Substitute in: ( )1 1y y m x x− = −

Parallel Line: 1) Use the SAME slope

2) Substitute in: ( )1 1y y m x x− = −

Perpendicular Line: 1) Use the OPPOSITE RECIPROCAL slope

(aka.. flip and change the sign)

2) Substitute in: ( )1 1y y m x x− = −

#12 – 15: Write an equation of the line with the given information. 12.Write the equation of the line with slope

2m = − and through the point 1 5( , )− .

13. Write the equation of the line through the points 4 5( , )− and 6 9( , )− .

14. Write the equation of the line parallel to 1

52

y x= − + through the point 4 5( , )− .

15. Write the equation of the line perpendicular

to 1

52

y x= − + through the point 4 5( , )− .

I can: Calculate the Line of Best Fit Using a Calculator

(turn on plot

and clear equations)

*Window (set window)

4

*Stat

*Edit (enter data)

*y

*Stat Calc

* # Lin Re g

=

→

#16 – 17: (a) Using the points provided, perform a regression to find the equation of the best fitting line. (b) Then find the indicated value. 16. (2,46), (5, 40), (7, 41), (12,32), (16, 28). Find y when x = 20.

17. (3,30), (4, 55), (8, 54), (11,60), (17, 65). Find y when x = 20.

Unit 2: Linear Systems

I can: Determine the number of solutions a system has from a….

Graph: 2 lines that are the same intersect at every point= infinitely many solutions 2 parallel lines that do not ever intersect =

No solutions

Equations: Any TRUE statement such as 5 = 5: infinitely many solutions Any FALSE statement such as 5 = – 2: No solutions

#18 – 20: Determine the number of solutions each system has.

18. 5 4 12

10 8 12

x y

x y

+ = −− − =

19. 4 3 6

3 9

x y

x y

− =+ =

20.

#21 – 22: Solve by graphing.

21. 6 9 27

4 3 9

x y

x y

− =+ =

22. 3 4 8

6 8 24

x y

x y

− − = −− − = −

#23 – 24: Solve by substitution.

23. 2 8

7 2 8

x y

x y

+ = −− + =

24. 3 9

2 2

x y

x y

− − = −+ = −

#25 – 26: Solve by linear combination/elimination.

25. 14 6 18

2 3 6

x y

x y

− − =+ =

26. 2 6 18

3 9 18

x y

x y

+ =− − = −

I can: Solve a system of equations in 3 variables….

*2nd, matrix, edit*3 (equations) x 4 (terms)*Enter coefficients*2nd, mode *2nd, matrix, math, B:, enter*2nd, matrix, 1:, enter

#27 – 30: Solve.

27.

6 3 2 1

3 6 5 10

5 2 2 3

x y z

x y z

x y z

− − + = −+ − =+ + =

28.

6 3 10

6 4 6 4

6 5 4

x y z

x y z

x y z

+ + = −+ − = −

+ + = −

9. A hotel has 260 rooms, some singles and some doubles. The singles cost $35 and the doubles cost $60. Because of a math teachers’ convention, all of the rooms are occupied. The sales for this particular night are $14,000. How many single rooms are there? How many double rooms are there?

30. A Broadway theater has 500 seats, divided into orchestra, main and balcony seating. Orchestra seats sell for $100, main seats for $75, and balcony seats for $50. If all the seats are sold, the revenue to the theater is $35,750. If all the main and orchestra seats are sold, but only half of the balcony seats are sold, the revenue is $31,750. How many seats of each kind are there?

I can: Determine a maximum or minimum from an objective quantity by evaluating the vertices…

1.) Graph the constraints 2.) Shade appropriately

3.) Use the vertices of the enclosed region to substitute into the objective quantity 4.) State the maximum or minimum value

31. Graph the constraints and use the vertices to find the maximum profit.

Objective Quantity Constraints P = 5x + 6y x ≥ 1

y ≥ 0 y + x ≤ 6

32. Graph the constraints and use the vertices to find the minimum cost. Objective Quantity Constraints

C=3x – 2y

0

0

10

2 3 12

2 8

x

y

y x

y x

y x

≥≥+ ≤+ ≥+ ≥

33. Write the objective quantity and the constraints for the scenario below.

An Algebra 2 quiz consists of computation problems and graphing problems. Computation problems are worth 6 points each and graphing problems are worth 10 points each. You can answer a computation problem in 2 minutes and a graphing problem in 4 minutes. You have forty minutes to take the quiz and may choose no more than 12 problems to answer. Assuming that you answer any problem you attempt correctly, how many of each type should you answer to get the highest score?

Unit 3: Absolute Value Equations, Inequalities, & Functions I can: Solve an absolute value equation or inequality.

Solve for x. 8 9x + =

Positive Case: Negative Case: x + 8 = 9 x + 8 = – 9 –8 –8 –8 –8 x = 1 x = –17 Solution(s): { }1 17,−

Solve for x and graph. 4 6 6x − ≥

greatOR than

4 6 6 4 6 6

6 6 6 6

4 0 4 12

4 4 4 4

0 3

x or x

x x

x x

− ≤ − − ≥+ + + +

≤ ≥

≤ ≥

Solution: ( ] [ )0 3, ,−∞ ∞∪

#34 – 39: Solve for x. Graph when a number line is provided.

34. 2 2 20x− − = 35. 5 10 55x+ =

36. 10 4 46x− = 37. 8 8 40x− − >

38. 2 6 8x − ≤ 39. 2 10 26x − <

Graph:  Graph: 

Graph: 

I can: Analyze the graph of an absolute value function when given an equation.

Determine all transformations for the equation:

13 2 1y x= − +

Vertical Shrink Right two up one

4 3 6y x= − + −

Vertical Stretch Left three Down six reflection

Determine the vertex, axis of symmetry and whether the function has a maximum or minimum.

13 2 1y x= − +

Vertex: ( 2 , 1 ) AOS: x = 2 Max or Min? Minimum

4 3 6y x= − + −

Vertex: (–3, –6 ) AOS: x = –3 Max or Min? Maximum

#40 – 43: Determine all transformations, vertices, axis of symmetry and whether the function has a maximum or minimum for the following. 40. 2 5 4y x= − + 41. 1

4 1 8y x= + −

Transformations: Transformations:

Vertex: Vertex:

AOS: AOS:

Max/Min: Max/Min:

42. 3 1y x= − + − 43. 3 6 2y x= − − +

Transformations: Transformations:

Vertex: Vertex:

AOS: AOS:

Max/Min: Max/Min:

Unit 4: Quadratic Functions #44 – 49: Solve using the algebraic method of your choice. Be sure to write your answers in set notation.

44. 26 9 132 0x x− − = 45. 27 44 12x x− = − 46. 24 6 6 0x x+ + =

47. 23 5 59x + = 48. ( )23 3 4 52x − − = − 49. 2 28 13 34 6 7x x x− + = − +

 

#50 – 51: Determine the average rate of change for each function for the given intervals.

50. ( ) 22 8 11f x x x= + + over the interval [ ]1 3, . 51. ( ) 22 8 11f x x x= + + over the interval [ ]2 3,− −

#52 – 54: Use the discriminant to determine the number and type of solutions for the equation. 52. 22 4 2 0x x+ + = 53. 2 8 2 9x x− − + = 54. 2 211 2 8 3 3x x x− = − + − #55 – 56: Write the equation or inequality for the following graphs. 55. 56. #57 – 58: Write the system of equations or inequalities for the following graphs. 57. 58.

Solution(s) a solution

 

 

2 4b ac−  

I can: Determine important pieces of a quadratic function and use them to graph.

Vertex Ordered Pair ( )2 sub in forba , y−

Axis of Symmetry Equation x = (comes from the x value from vertex) Maximum or Minimum Equation y = (comes from the y value from vertex)

Negative leading coefficient → Maximum Positive leading coefficient → Minimum

y – intercept Ordered Pair ( )sub in0 for solve forx , y

( )0, y

#59 – 61: Graph each function & identify the important information.

59. 2 10 25y x x= + + Vertex:

Axis of symmetry:

y - intercept:

CIRCLE ONE: Maximum or Minimum

60. 22 4 1y x x= − − + Vertex:

Axis of symmetry:

y - intercept:

CIRCLE ONE: Maximum or Minimum

61. 23 1 2y ( x )= − + − Vertex:

Axis of symmetry:

y - intercept:

CIRCLE ONE: Maximum or Minimum

#62-63: Identify the key features of the graphs. 

62. 63. Domain: Domain:

Range: Range:

Increase: Increase:

Decrease: Decrease:

( ) 0f x > : ( ) 0f x > :

( ) 0f x < : ( ) 0f x < : .

Solutions: Solutions:

Vertex: Vertex:

MAX or MIN MAX or MIN

#64 – 65: Complete the square to write the equation in vertex form.

64. ( ) 2 12 32f x x x= + + 65. ( ) 22 20 44f x x x=− + −

66. The formula 216 48 135h( t ) t t= − + + gives the height of a ball thrown from the top of a building with an initial speed of 48 ft/sec, where t is measured in seconds.

a.) What is the y-intercept? What does it represent?

b.) When will the ball reach 160 feet?

c.) Will the ball ever reach 180 feet? Explain.

d.) How long does it take for the ball to reach its maximum height?

e.) How many seconds after its release will the ball hit the ground?

For your use:

 

 

y

x

Unit 5 I can: Determine the end behavior of a polynomial. Describe the end behavior of the graph of

5 3( ) 2 4 5 4f x x x x= − + − .

( )as , x f x→−∞ →−∞

( )as , x f x→∞ →∞

Describe the end behavior of the graph of 2 4( ) 4 4 1f x x x x= − − − + .

( )as , x f x→−∞ →−∞

( )as , x f x→∞ →−∞

#1 – 4: Describe the end behavior of the graph of each polynomial. 4.) 3( ) 8 1 2f x x x= − − 2.) 5 2( ) 5 2 1f x x x= + − 3.) 2( ) 2 4f x x x= − + 4.)

2 6 5( ) 4 11 5f x x x x= + −

I can: Find important pieces of a polynomial function from a graph. Maximum/Minimum x – intercepts

                                                                                        Press:    

 2nd, Trace   Select either:     3: minimum or     4: maximum   

Use your cursor to go to the left of the vertex, press enter, then the right, press enter, then enter 

one more time 

Press:      2nd, Trace   Select:     2: zero       

Use your cursor to go to the left of the x ‐ intercept, press enter, then the right, press enter, then 

enter one more time 

#5 – 6: Determine all of the important pieces of the polynomial function.

5.) 4 2f(x) x 4x 3 4x= − + − − Domain: Range:

Local Maxima: Local Minima:

Increasing: Decreasing:

x – intercept(s): y – intercept:

( ) 0f x > :

( ) 0f x < :

Left End Behavior:

Right End Behavior:

6.) 3 2f (x ) x 4x 2= − + −

Domain: Range:

Local Maxima: Local Minima:

Increasing: Decreasing:

x – intercept(s): y – intercept:

( ) 0f x > :

( ) 0f x < :

Left End Behavior:

Right End Behavior:

Left side 

right side 

Odd degree

Positive leading coefficient 

Left side

right side

Even degree

Negative leading coefficient 

I can factor a polynomial with a higher degree. Quartic Trinomial Polynomial with an even number of terms

4 22 24 0x x+ − =

What two numbers add to be 2 and multiply to be

24− ?

6 and 4−

( ) ( )2 26 4 0x x+ − =  

3 20 6 42 7x x x= + + +

Group: ( ) ( )3 20 6 42 7x x x= + + +

GCF: ( ) ( )20 6 7 1 7x x x= + + +

Regroup: ( ) ( )20 6 1 7x x= + +

#7 – 12: Factor.

7.) 4 20 2 6 20x x= + − 8.) 3 20 5 3 25 15x x x= − − + 9.) 4 20 2 14 36x x= + −

10.) 3 20 3 12 5 20x x x= + − − 11.) 3 20 6 9 4 6x x x= − + − 12.)

4 20 3 6 3x x= + +

I can: Describe the graph of a polynomial in as many ways as possible.

This graph has two turns, so the degree is probably cubic. The leading coefficient is positive because the right end rises. The degree is odd because the left end falls (is the opposite of the right). If the degree is 3, there are three zeros. One is a positive real zero (x – intercept). The other two will be imaginary. The y – intercept is negative.

#13 – 15: Describe the graph in as many ways as possible.

13.) 14.) 15.)

I can: Divide a polynomial by another polynomial.

Long division Synthetic division 3 2(6 21 9 9) ( 2)x x x x− + + ÷ −

2

3 2

3 2

2

2

6 9 9 2 6 21 9 9

6 12

9 9

9 18

9 9

9 18

9

x xx x x x

x x

x x

x x

x

x

− −− − + +

−

− +

− +− +− +

−

2 926 9 9 xx x −− − −  

3 2(6 21 9 9) ( 2)x x x x− + + ÷ −

2 0

2

x

x

− ==

2 6 21 9 9

12 18 18

6 9 9 9

−− −

− − −

2 926 9 9 xx x −− − −

#16 – 19: Divide using the method of your choice. 16.) 3 2( 2 4 8) ( 2)x x x x+ − + ÷ + 17.) 3 2(2 9 15) (2 5)x x x− + ÷ −

18.) 2( 7 18) ( 2)x x x− − ÷ + 19.) 3 2(3 5 10 3) (3 1)x x x x− + − ÷ +

I can: Find the remainder when dividing polynomials. Using subtitution By evaluating

Find the remainder when 3 26 21 9 9x x x− + + is

divided by 2x −

2 0

2

x

x

− ==

2 6 21 9 9

12 18 18

6 9 9 9

−− −

− − −

 

Find the remainder when 3 26 21 9 9x x x− + + is

divided by 2x −

3 2(2) 6(2) 21(2) 9(2) 9 9f = − + + = −

#20 – 21: Find the remainder.

20.) When 3 32 6 12 5x x x+ − + is divided by 1x − . 21.) When 23 13 74 6 3x x x− − + − is divided by

1x + .

I can find all zeros of a polynomial,

given one zero.

If 3 is a zero of 3 2( ) 9 9f x x x x= + − − , find the

other zeros.

3 1 1 9 9

3 12 9

1 4 3 0

− −

2( 3)( 4 3) 0

( 3)( 3)( 1) 0

3, 3, 1

x x x

x x x

x

− + + =− + + =

= − −

If 2 is a zero of 3 2( ) 2 10 4f x x x x= + − + , find the

other zeros.

2 1 2 10 4

2 8 4

1 4 2 0

−−

−

2( 2)( 4 2) 0x x x− + − =

4 16 4( 2) 4 24 4 2 62 6

2 2 2x

− ± − − − ± − ±= = = = − ±

2 6, 2x =− ±

#22 – 25: Find all zeros of the polynomial, if one is given.

22) 3 2( ) 2 9 18f x x x x= + + + ; -2 23) 3 2( ) 7 4 12f x x x x= + + − ; -6 24) 3 2( ) 2 3 23 12f x x x x= + − − ; 12−

I can: Write a polynomial function of least degree given zeros. Write the polynomial function of least degree that has zeros of 5, 2 and 1− .

2

2

3 2 2

3 2

( 5)( 2)( 1) ( )

( 2 5 10)( 1) ( )

( 3 10)( 1) ( )

3 3 10 10 ( )

2 13 10 ( )

x x x f x

x x x x f x

x x x f x

x x x x x f x

x x x f x

+ − − =

− + − − =

+ − − =

− + − − + =

+ − + =

Write the polynomial function of least degree that has zeros of 4, 2 and 2i i− − .

2 2

2

3 2

( 4)( 2 )( 2 ) ( )

( 4)( 2 2 4 ) ( )

( 4)( 4) ( )

4 4 16 ( )

x x i x i f x

x x i i i f x

x x f x

x x x f x

+ − + =

+ + − − =

+ + =

+ + + =

#25 – 27: Write a polynomial function of least degree that has the following zeros.

25.) 4, 1, 6 26.) 0, 3 , 3i i− 2, -1 and 0 35.) 12 , 4 , 4i i− −

1.) Use your graphing calculator to find one zero. 2.) Use synthetic division to factor the polynomial. 3.) Factor completely set factors equal to zero to solve. 4.) If the resulting polynomial is not factorable, use the quadratic formula!