Final Exam Review Algebra 2 2015-2016 Semester 1
Name:______________________________
Module 1
Find the inverse of each function.
1. f  x   10  4 x
2.
g  x   15x  10
Use compositions to check if the two functions are inverses.
3. s  x   7  2x and 𝑡(𝑥) = − 2 𝑥 + 2
1
7
4.
hx 
1
x  4 and j  x   3 x  12
3
Find the inverse of each function. Then graph the function and its inverse.
5. f  x   5 x  10
f 1  x   _________________
6. f  x  
9
x 5
2
f 1  x   _________________
Determine whether the inverse of each relation is a function. If so, state whether the inverse is a one-to-one or
a many-to-one function. If not, explain the reasoning.
7.
8.
Let g(x) be the transformation of f(x). Write the rule for g(x) using the change described.
9. Reflection across the y-axis followed by a vertical shift up 3 units.
10. Horizontal shift right 2 units and a vertical stretch by a factor of 3
11. Vertical compression by a factor of
1
followed by a vertical shift down 6 units
8
12. Reflection across the x-axis, followed by a vertical stretch by a factor of 3, a horizontal shift 6 units left, and a vertical
shift 4 units down.
13. Find the equation of the parabola in vertex form (𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘).
14. Write the given interval in inequality, set, and interval notation.
Inequality: _________________
Set Notation: _________________
Interval Notation: _________________
Use the given graph to answer the following questions #15-17.
15.On which intervals is the function increasing and
decreasing?
16. What are the local maximum and minimum values?
Max: _____________
Min: ______________
17. What are the zeros of the function?
18. In linear regression, what does the r value stand for?
19. Use the information in the following table to make a model for the data.
Hours studying
2
8
7
4
3.5
Grade on exam
37
98
87
70
68
a) Using your graphing calculator, find a linear model that fits this data. Round to two decimal places.
b) Use this model to predict a student’s grade if they studied for 6 hours.
Module 2
Use the graph below for 1–4.
1. Write a function in standard form to match the graph. Let b =1.
2. Find the vertex of the function.
3. Find the domain of the function.
4. Find the range of the function.
5. Given that 𝑓(𝑥) = |𝑥 − 4| + 3 determine if each statement is True or False.
A The domain is all real numbers.
True
False
B The vertex is (-4, 3).
True
False
C The vertex is (4, 3).
True
False
D The range is all real numbers.
True
False
6. If 𝑔(𝑥) = |𝑥 + 5| write a function h(x) that is g(x) translated down 2 units and vertically stretched by a factor of 2.
7.Does each number line show a solution to 2 x  7  4?
A
Yes
No
Yes
No
Yes
No
B
C
8.Solve 4|3𝑥 + 5| ≥ 16
9. Solve 4|2𝑥 − 1| + 3 = 2.
Module 3
1. A bridge is 72 feet above a river. How many seconds does it take a rock dropped from the bridge to pass by a
tree limb that is 5 feet above the water? (Use the model h(t) = h0 -16t 2 .)
2. The area of the rectangle in the figure is 42 square meters. What are the width and length of the rectangle? Give
answers both exactly and approximately (to the nearest tenth).
3. Solve the equation 2x2 + 3 = 5 using the indicated method.
A. Solve by graphing
x
f(x)
-2
-1
0
1
2
B. Solve by taking square roots
4. Which of the following results in a number with both a real and an imaginary part?
A. (3 +2i)(3 - 2i)
B. (5 - 2i)(3 + 4i)
C. (  4i)  (  5i)
D. (9  8i)  (4  8i)
5. Determine if each quadratic equation has real solutions. Choose Yes or No.
A. 12x 2  5  3
B.
3x 2  1  8x 2  9
C. 4(3x 2  7)  3( x 2  5)
Yes
No
Yes
No
Yes
No
6. A tennis ball is hit with an initial vertical velocity of 60 ft/s. The function h(t)  16t 2  60t  2 models the height h
(in feet) of the tennis ball at time t (in seconds). Does the ball reach each of the following heights? Choose Yes or No.
A 44 feet
Yes
No
B 52 feet
Yes
No
C 60 feet
Yes
No
7. Determine the number and types of solutions for each quadratic equation.
A. 9x2  7x  6
Number: ____ Type: _________
B. 3(x2  4) 2x2  3
Number: ___ Type: ________
C. 5 + 3x2 = 7x2 - 10
Number: ____ Type: _______
For 8–11, perform the indicated operation
8. (4  11i)(5  8i)
9. (6  3i)(2i  4)  (4  5i)
10. (2  3i )(2  3i )
11. (3i – 4) + (17i + 12)
Solve the quadratic equation by completing the square.
12. 0  x 2  16x  33
For 13–15, solve the quadratic equation by taking square roots.
13. 5x2 = 81
14. 2x2  30  0
15. 3x2  27  0
For 16–17, use the quadratic formula to determine the solutions to each equation.
16. 3x2 7x  9  0
17. 5x2  9x  8  0
18. The graph of ƒ (x) =
1 2
x  3x is shown. Use the graph to determine how many real solutions the following
2
equations have:
A.
1 2
9
x  3x   0
2
2
B.
1 2
x  3x  3  0
2
C.
1 2
x  3x  6  0
2
Module 4
1. Cara and Ellen both take a walk through the park. Cara follows a path modeled
2
by the equation y x
y
x
point where the paths will cross. Choose Yes or No.
A. (4, 7)
Yes
No
B. (2, 5)
Yes
No
C. (3, 6)
Yes
No
D. (1, 4)
Yes
No
2. What is the equation of a parabola with a vertex at (0, 0), a focus at (5, 0), and a directrix x -5?
3. What is the equation of the parabola with the focus (2, 4) and the directrix x  2?
4. Find the missing characteristics of the parabola from the given information.
Focus: (0,7) directrix: y = -7
Vertex: __________
Opens: __________
p: _________
Equation: ______________________
5. Which statement about a parabola is
not true?
A. The focus of the parabola lies on the axis of symmetry.
B. The directrix of the parabola is parallel to the axis of symmetry.
C. Each point on the parabola is equidistant from the directrix and the focus.
6. Which line intersects the parabola shown below in exactly two points?
A.
x  2
B.
y  x2
C.
y  x  2
7. Find the equation of the parabola with the focus ( 2, 2) and the directrix y
𝑥 2 + 𝑦 = 16
8. Solve the system {
𝑦+𝑥 =4
8 x  y 2  12
9. Solve the System 
x  y  3
10. Choose True or False for each statement about the parabola with the equation y 2  12x  2y  23.
A. The vertex is (1, 2).
True
False
B. The focus is (1, 1).
True
False
C. The directrix is x  5.
True
False
D. The parabola opens to the left.
True
False
11. Determine if each line intersects the parabola shown below in exactly two points. Choose Yes or No.
A.
y 2
Yes
No
B.
x 2
Yes
No
C.
y  2
Yes
No
D.
xy 2
Yes
No
E.
x  y  2
Yes
No
12. The equation for the cross section of a parabolic telescope mirror is 34𝑦 = 𝑥 2 measured in centimeters. How far is
the focus from the vertex of the cross section?
Module 5
1.
Give the coordinates of the transformed reference points if 𝑓(𝑥) = 𝑥 3 is stretched vertically by a factor of 3 and
then translated 5 units right and 2 units up.
1
2. Draw the graph if the function shown is transformed to 𝑔(𝑥) = 4 (𝑥 + 5)3 − 3. Draw the new curve and give the
coordinates of the transformed reference points.
3. Write the equation of the function the graph represents. Use the form g ( x)  a  x  h   k .
3
Equation: ________________________
4. Write the equation of the function the graph represents. Use the form
g ( x)  a  x  h   k .
3
Equation: ________________________
5. Write a function that transforms f ( x)  x3 with a vertical stretch by a factor of 3, a left shift of 4 units and a shift up
7 units.
6. Describe how the graph of g ( x) 
1
3
 x  7   4 is related to the graph of f(x) = x3. (4 pts)
2
7. Describe how the graph of 𝑔(𝑥) = −64(𝑥 − 3)3 − 10 is related to the graph of f(x) = x3. (4 pts)
8. Sketch the graph of the function 𝑓(𝑥) = 𝑥(𝑥 − 1)2 and determine the following.
Domain: ______________
Range: _______________
End Behavior: _____________________________
_____________________________
x-Intercepts: ________________________________
9. Sketch the graph of the function 𝑓(𝑥) = (𝑥 + 1)(𝑥 − 2)(𝑥 + 3)and determine the following.
Domain: ______________
Range: _______________
End Behavior: _____________________________
_____________________________
x-Intercepts: ________________________________
10. Which of the following is a true statement about the graph of p( x)  x  x  4  x  2  ?
2
A. The graph crosses the x-axis four times and is never tangent to the x-axis.
B. The graph crosses the x-axis three times and is never tangent to the x-axis.
C. The graph crosses the x-axis two times and is tangent to the x-axis once.
D. The graph crosses the x-axis three times and is tangent to the x-axis once.
11. Which of the following statements are true about the polynomial function p ( x ) ? (The zeros of p ( x ) are integers,
and the graph of p ( x ) does not cross the x-axis at places other than those shown.)
A. The degree of p(x) is even
B. The leading coefficient of p(x) is negative
C. P(x) has 6 distinct zeroes
D. P(x) has a 1 global minimum
For number 12-13, determine the number of turning points and number and type (global or local) of any maximum and
minimum values.
13. 𝑓(𝑥) = (𝑥 − 1)3 (𝑥 + 5)
12. 𝑓(𝑥) = 𝑥(𝑥 + 4)(𝑥 − 1)
Turning points: _______________
Turning points: _______________
Maximum: _________________________
Maximum: _________________________
Minimum: __________________________
Minimum: __________________________
14. An x by x square is cut from each corner of a rectangular sheet of cardboard that is 7 inches wide and 10 inches long.
The sides are then folded up to form a box. Write the equation that represents the volume of the box. To the
nearest tenth, what value of x maximizes the volume of the box?
15. Which is a graph of an even function with a positive leading coefficient?
a.
B.
C.
D.
16. The base of a right rectangular pyramid has a length of (12 – w) and a width of w centimeters. The height of the
2
pyramid is (12 − 𝑤). What is the maximum volume of the pyramid?
5
1


 The formula for the volume of a pyramid is V  LWH 
3


Module 6
A)

 a  b  a  ab  b 
B)
C)

1. Does each product result in a 3  b3 when simplified?
2
2
Yes
No
 a  b  a2  ab  b2 
Yes
No
 a  b  a  b a  b 
Yes
No
For 2-6, Simplify the polynomial, write in standard form, identify the degree and the leading coefficient.
2.
(4𝑥 5 − 7𝑥 3 + 2) + (−2𝑥 5 − 4𝑥 3 + 4)
3. (5𝑥 3 − 7𝑥 2 + 3) − (7𝑥 3 + 2𝑥 2 − 4)
4. (8𝑥 3 − 2𝑥 2 − 𝑥 + 4) − (𝑥 3 + 5𝑥 2 − 5𝑥 − 1) + (𝑥 2 − 4)
5.
(4𝑥 − 1)(𝑥 2 + 2𝑥 − 3)
6. (3𝑥 2 − 2𝑥)(𝑥 2 + 3𝑥 − 2)
7. A rectangular yard has a length of (𝑥 2 − 13𝑥 − 3) yards and a width of x yards. Write the simplified polynomial that
represents the perimeter of the yard. Use the polynomial to estimate the perimeter if the width is 18 yards.
8. A biologist has found that the number of branches on a certain rare tree in its first few years of life can be modeled
by the polynomial b( y)  4 y 2  y . The number of leaves on each branch can be modeled by the polynomial
L( y)  2 y 3  3 y 2  y where y is the number of years after the tree reaches a height of 6 feet. Write a polynomial
describing the total number of leaves on the tree.
9. If 𝑇(𝑥) = 2.3𝑥 2 − 46.3𝑥 + 23.9 represents the number of teenagers at a concert, and
𝐴(𝑥) = 4.8𝑥 2 + 24.8𝑥 + 45.8 represents the number of adults at a concert, write a polynomial function to
represent the total number of people at the concert.
For 10-17, factor the polynomial completely, or identify it as irreducible.
11. 2𝑥 2 − 2𝑥
12. 2𝑏 3 + 2𝑏 2 − 180𝑏
13. 27𝑡 3 − 64
14. 4𝑥 3 + 2𝑥 2 − 2𝑥 − 1
15. 9𝑓 3 + 36𝑓 2 − 4𝑓 − 16
16. 𝑥 2 − 10𝑥 + 25
17. 2𝑟 2 − 12𝑟 − 80
18. 4𝑥 4 + 32𝑥
19. 81𝑥 2 + 49
20. The voltage generated by an electrical circuit changes over time according to the polynomial
𝑽 = 𝒕𝟑 − 𝟏𝟎𝒕𝟐 + 𝟒𝒕 − 𝟒𝟎 where V is in volts and t is in seconds. Factor the polynomial to find the times when the
voltage is equal to zero.