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2015 Math III End of Year Review
Linear
Write the slope-intercept form of the equation.
1.
Verbal Description
3. When x is increased by twice
y, the result is 4.
4. The opposite of half the value
of x, when increased by 1
results in the value of y.
Absolute Values
5. 𝑦 = |𝑥 + 1|
6. 𝑦 = −|−𝑥 − 2| + 1
Symbolic
2. through (−2, −3); slope = 4
Graph
Piecewise
𝑓(𝑥) = {
𝑥 + 3,
2𝑥,
7. 𝑓(2)
𝑖𝑓 𝑥 ≤ 0
𝑖𝑓 𝑥 > 0
8. 𝑓(−4)
9. 𝑓(0)
1
10. 𝑓(2)
11. Graph
Quadratic
12. A) What is the factored form of 𝑥 2 + 9? 12.B) What about factored form of 18x2 +27x +10
Composition of Functions
You have a coupon for 20% off a meal at your favorite restaurant. You also have a $15 gift
card to the restaurant.
13. Write a function representing the cost, x, of the meal just with the coupon.
14. Write a function representing the cost, x, of the meal just with the gift card.
15. Does it save you more money if the coupon is applied first or if the gift card is applied
first? How much is the savings?
Polynomials
16. Look at the expression 3𝒙𝟐 − 𝟐𝒚 + 𝟓 + 𝟒(𝒙 − 𝟏) to identify its elements.
Coefficients:
Constant:
Factor:
Terms:
17. An equation is shown 𝑓(𝑡 2 + 1) ∙ 𝑥 = 𝑥(𝑡 3 − 𝑡 2 + 2𝑡 − 1) Does the
function 𝑓 depend on the variable 𝑥?
a. Yes, the function 𝑓 is dependent on 𝑥.
b. No, the function 𝑓 is not dependent on 𝑥.
c. It is impossible to determine whether or not 𝑓 is dependent on 𝑥.
18. The polynomial 𝑝(𝑥) contains the points (3,0) and (8,1).
Which statement is true according to the Remainder Theorem?
a. (𝑥 + 3) is a factor of 𝑝(𝑥).
b. (𝑥 − 3) is a factor of 𝑝(𝑥).
c. (𝑥 + 8) is a factor of 𝑝(𝑥).
d. (𝑥 − 8) is a factor of 𝑝(𝑥).
Simplify the following polynomials.
19. (8𝑥 2 + 3𝑥 + 4) + (5𝑥 2 − 6𝑥 − 8)
20. (5𝑥 2 − 6𝑥 + 1) − (𝑥 2 − 5𝑥 + 3)
21. 2𝑥(𝑥 2 − 𝑥 + 5) − 5𝑥(3𝑥 − 2)
22. (2𝑘 + 1)(𝑘 2 + 7𝑘 − 9)
23. Fluffy Pet Products sells a pet toy that brings in revenue represented by the function
𝑟(𝑥) = −0.05𝑥 2 + 100𝑥 + 100, where x is the number of pet toys. The production cost
for the pet toy is represented by the function 𝑐(𝑥) = 0.42𝑥 + 50.
Which function represents the profit Fluffy Pet Products earns on the pet toy?
a.
b.
c.
d.
0.05𝑥 2 − 99.58𝑥 − 50
−0.47𝑥 2 + 50𝑥 + 100
−0.05𝑥 2 + 99.58𝑥 + 50
−0.05𝑥 2 + 99.58𝑥 + 150
Find the quotient and the remainder and state if the given divisor is a factor of the
polynomial.
24. (𝑝4 − 1) ÷ (𝑝 − 1)
25. Use the graph below to answer the following questions:
a. At what x-value does the function have a relative maximum?
b. At what x-value does the function have a relative minimum?
c. Along what interval is the function decreasing?
d. Along what intervals is the function increasing?
e. Is the degree of this function even or odd? How can you tell?
f. Is the leading coefficient of this function positive or negative? How can you tell?
26. Graph the following: 𝑓(𝑥) = (𝑥 − 4)2 (𝑥 + 2)
27. How many solutions to the equation 𝑥 3 − 1 = 0 are imaginary?
28. Which function is an even function?
a. 𝑓(𝑥) = 4𝑥
b. 𝑓(𝑥) = 𝑥 2 + 2𝑥 + 1
c. 𝑓(𝑥) = 𝑥 3 + 𝑥
d. 𝑓(𝑥) = 5𝑥 4 + 3𝑥 2 + 1
Average Rate of Change
29. The table represents the number of people who held bookstore memberships in 2005 and
2008.
Years Number of Members
2005
330,000
2008
480,000
What is the average rate of change per year?
a. 37,500 per year
b. 50,000 per year
c. 150,000 per year
d. 270,000 per year
Inverse Functions
30. What is the inverse of the function 𝑓(𝑥) = 4𝑥 2 for 𝑥 ≥ 0?
1
a. 𝑓 −1 (𝑥) = 2 √𝑥
𝑥
b. 𝑓 −1 (𝑥) = 4
𝑥
c. 𝑓 −1 (𝑥) = 2
1
d. 𝑓 −1 (𝑥) = ± 2 √𝑥
Radicals
Sketch the graph and state the domain and range.
31. 𝒇(𝒙) = −√𝒙 − 𝟑 + 𝟏
32. Solve the equation √𝒙 + 𝟓 − 𝟏𝟎 = −𝟖
33. Solve the equation √𝟏𝟔 − 𝒙 = −𝟒
Rational
𝟗𝒙
34. Simplify 𝒙+𝟓 ÷
𝟔
(𝒙−𝟗)(𝒙+𝟓)
(𝒙+𝟓)𝟐
𝒙
35. Simplify 𝒙+𝟏 − 𝟐𝒙+𝟐
36. Solve
𝒙𝟐 −𝟒𝒙−𝟐𝟏
𝒙𝟐 −𝒙−𝟓𝟔
𝒙−𝟑
= 𝒙𝟐 −𝒙−𝟓𝟔 + 𝟏
−𝟐
37. Identify the vertical asymptotes of 𝒇(𝒙) = 𝒙𝟐 −𝟔𝒙+𝟖
38. Graph 𝒇(𝒙) =
𝟏
𝒙−𝟏
+𝟑
Trigonometry
39. Solve the triangle and find the area.
40. Solve the triangle and find the area.
41. Identify the function graphed below.
a. sin(2𝑥)
1
b. sin (2 𝑥)
c. 2 cos(𝑥)
d. cos(2𝑥)
5𝜋
42. What is the value of sin ( 2 )?
1
𝜋
43. Identify the amplitude, period, midline, and phase shift 𝑓(𝑥) = 2 sin[4 (𝑥 − 4 )] − 1
Logarithmic
44. Which represents the solution to 5 ∙ 32𝑥 = 20?
a. log 3 4
b. 2 log 3 2
1
c. 2 log 4
d.
1
2
log 3 4
45. The population growth of a city can be modeled by the equation 𝑃 = 130,000 𝑒 0.09𝑡 ,
where 𝑃 is the population in thousands and 𝑡 is years since 1995.
In what year does the model predict that the city reaches a population of approximately
600,000 people?
a. 2012
b. 2014
c. 2016
d. 2017
46. Graph and identify the domain and range 𝑓(𝑥) = log(𝑥) + 1
Expand each logarithm.
𝑥2
47. log 8 (𝑦 6 )
48. ln(𝑥√𝑦)
Condense each expression to a single logarithm.
log 6
50. ln 8 − 2 ln 2
49. 2 log 3 7 + 33
51. Sally currently pays a $3,034 premium for health insurance. If the premium increases at
an annual rate of 9.2% per year, how many years will it take for the premium to be
$7,315.47?
Critical Thinking
52. The sum of Vince’s age and Emy’s age is 34. Emy is 2 years less than twice Vince’s age.
Which equation represents this situation if 𝑣 represents Vince’s age?
a. (2 − 2𝑣) + 𝑣 = 34
b. (2 − 2𝑣) − 𝑣 = 34
c. (2𝑣 − 2) + 𝑣 = 34
d. (2𝑣 − 2) − 𝑣 = 34
53. An equation is shown. −(56𝑦 3 𝑥 4 )0 + 3𝑥 + 4 = 𝑥 + 𝑦
Solve for 𝑦.
54. A function is shown. Which word describes the graph of this function?
52 ∙ 𝑥 3 ∙ (3𝑥 2 )3
𝑦=
125 ∙ 𝑥 6 ∙ (2𝑥)2
a. Exponential
b. Linear
c. Logarithmic
d. Quadratic
55. An employer uses a function to calculate each employee’s salary. The input is the number
of hours and the output is the salary. Since the employees’ time clock is in 15-minute
intervals, what is a good domain for this function?
1
a. Non-negative multiples of 4
b. Non-negative rational numbers
c. Non-negative multiples of 15
d. Non-negative integers