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QCC MA-119 Final Review Questions.

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 1 of 12

Graphs and Functions

1. Use the graph of the function, ? = ?(?), to answer the question.

a) Estimate ?(1)

Answer: ?(1) ≈ 5 (find a point on the graph that has ?-coordinate = 1).

b) Estimate the solution(s) to ?(?) = 9

Answer: ? ≈ 1.75 (find all points on the graph that have ?-coordinate=9; in this case,

there's just one such point).

c) Estimate the coordinates of all intercepts.

Answer: ?-intercept is approximately(−8.5,0). (All ?-intercepts have ?-coordinate=

0.) There’s only one ?-intercept in this graph; ?-intercept is approximately(0,3). (All ?-

intercepts have ?-coordinate= 0. There’s only one y-intercept in this graph.)

d) Estimate the domain and range of the function

Answer: Domain: (−∞, ∞). (The arrows indicate that the graph extends, without

breaks, indefinitely to the left and indefinitely to the right.) Range: (−∞, ∞). (The

arrows indicate the graph extends, without breaks, indefinitely up and indefinitely

down.)

e) Estimate the minimum or maximum ?-value

Answer: no minimum ?-value and no maximum ?-value. (The arrows indicate that the

graph extends indefinitely up and indefinitely down.)

f) Estimate the ?-value at which the minimum or maximum is achieved?

Answer: no such ?-values because of the answer to part e).

g) Estimate the values of ? for which ? < 0

Answer: (−∞, 8.5) For these ?-values the graph is below the ?-axis so for these ?-

values, ?<0. See how this answer relates to part c)

h) When 1 < ? < 2, is the function increasing or decreasing?

Answer: Increasing. Move to the right within the stated interval of ?-values. The graph

is going upwards for those ?-values.

2. Use the graph of the function, ? = ?(?), to answer the question.

a) Estimate ?(−11)

Answer: ? = ?(−11) ≈ 7

b) Estimate the solution(s) to ?(?) = −4

Answer: ? ≈ −13 and ? ≈ −6

c) Estimate the coordinates of all intercepts.

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 2 of 12

Answer: ?-intercepts: (−14,0), (0,0), (9,0); ?-intercept: (0,0)

d) Estimate the domain and range of the function

Answer: Domain (−∞, ∞) Range [−7.75, ∞)

e) Estimate the minimum or maximum ?-value

Answer: minimum ?-value ≈ −7.75; no maximum ?-value

f) Estimate the x-value at which the minimum or maximum is achieved?

Answer: For minimum: ? ≈ −10 (The ?-coordinate of the point with the minimum ?-

value); For maximum: no such ?-value

g) Estimate the values of ? for which ? < 0

Answer: (−14,0) ∪ (0,9) (the symbol ∪ stands for union or “together with”. The

endpoints are not included because there we have ? = 0 not ? < 0

h) When 4 < ? < 5, is the function increasing or decreasing?

Answer: decreasing

3. Compute

?(4)−?(1)

4−1

for the given function.

a) ?(?) = ?2 − 3? + 1

Answer: 2

b) ?(?) = √?

Answer:

1

3

Linear Functions

4. Suppose that a manufacturer wishes to use a linear function to model how the demand for its

product affects the product’s price. The unit price function has the form ?(?) = ?? + ? where

? is the number of units of the product in demand. Suppose that when 2400 units are

demanded then the unit price is $250 and when 3000 units are demanded then the unit price

is $300.

Number of units ? Price per Unit ?

XXXXXXXXXX

XXXXXXXXXX

a) Determine the equation of the function ?(?) = ?? + ?

b) According to this model, what is the unit price ?(?) if the demand is ? = 3600?

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 3 of 12

Answer: a) ?(?) =

1

12

? + 50 b) $350

5. A linear function ? passes through (−1,5) and (2, −4). Find the function value ?(−2).

Answer: ?(−2) = 8

6. Markos currently has 200 songs in his music collection. Every month, he adds 15 new songs.

Write a formula for the number of songs, ?, in his collection as a function of time, ?, the

number of months. How many songs will he own in a year?

Answer: ?(?) = 15? + 200; 380 songs

7. A town’s population has been growing linearly. In 2004 the population was 6,200. By 2009

the population had grown to 8,100. Assume this trend continues.

a) Write a linear equation to model the population growth.

Answer: ?(?) = 380? + 6200, where ? is the number of years since 2004.

b) Predict the population in 2013.

Answer: 9,620

c) Identify the year in which the population will reach 15,000.

Answer: Sometime around the year 2027

Quadratic Equations (Factoring and Quadratic Formula)

8. Factor completely. 40?4 + 22?3 − 6?2

Answer: 2?2(4? + 3)(5? − 1)

9. Solve the quadratic equations

a) 4?2 + ? = 3

Answer: ? = −1 or ? = −

3

4

b) (? + 1)(? − 3) = 2

Answer: ? = 1 ± √6

c) ?2 + 7? + 12 = 0

Answer: ? = −3 or ? = −4

d) 2?2 + 1 = ?

Answer: ? =

1±?√?

4

e) 2(? XXXXXXXXXX = 0

Answer: ? = −3 ± 2?

f)

1

3

?2 + ? = −

1

2

Answer: ? =

−3±√3

2

g) 4?2 − ? − 3 = 0

Answer: ? = 1 or ? = −

3

4

h) (? −

1

3

)

2

−

4

9

= 0

Answer: ? = 1 or ? = −

1

3

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 4 of 12

Quadratic Functions

10. Graph the following quadratic functions.

a) ?(?) = −?2 − 6? − 5

b) ℎ(?) = −(? − 3)(? + 2)

c) ?(?) = 2?2 − 3? − 2

d) ?(?) = −2?2 − 4? + 5

e) ?(?) = (2? − 1)(? + 3)

f) ?(?) = 2?2 + 8

Before graphing the parabolas, state which open upwards and which open downwards.

For each parabola include

a) The coordinates of all intercepts

b) The coordinates of the vertex

c) The equation and graph of the axis of symmetry

d) The domain and range in interval notation

e) The coordinates of an additional point on the graph

f) The maximum or minimum range value

g) The domain value at which the max or min is reached

h) ?-interval(s) at which the function is negative

i) ?-interval(s) at which the function is positive

Application of Quadratic Equations/Functions

11. The length of a rectangle exceeds twice its width by 3 inches. If the area is 10 square inches,

find the rectangle's dimensions. Round to the nearest tenth of an inch.

Answer: Approximately 1.6 in by 6.2 in.

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 5 of 12

12. The hypotenuse of a right triangle is 6 feet long. One leg is 2 feet shorter than the other. Find

the lengths of the legs. Round to the nearest tenth of a foot.

Answer: Approximately 5.1 ft and 3.1 ft.

13. A retailer estimates that, by charging x dollars each for a particular phone case, she can sell

40 − ? units each week. The function ?(?) = ?(40 − ?), models the weekly revenue, ?(?),

received when the selling price is ?.

a) Interpret (0, ?(0)) and (1, ?(1)) in context.

Answer: When the charge is $0, the revenue is $0; When the charge is $1, the revenue

is $39

b) Find the selling price that will give the maximum revenue, and then find the amount of

the maximum revenue.

Answer: The maximum revenue is $400 when the charge is $20

c) For which selling price(s) will the weekly revenue equal $300?

Answer: $10 and $30

14. A ball is thrown vertically upward, at the rate of 120 ft/sec, from a rooftop with a height of 50

feet. The quadratic function ℎ(?) = −16?2 + 120? + 50 models the height of the ball, from the

ground, at time ?.

a) Interpret (0, ℎ(0)) and (1, ℎ(2)) in context.

Answer: 0 seconds after the ball is thrown the ball is 50 feet from the ground; 1

second after the ball is thrown the ball is 154 feet from the ground.

b) Find how long it will take the ball to reach its maximum height, and then find the

maximum height.

Answer: The maximum height of 275 feet is reached in 3.75 seconds

c) At which time(s) does the ball reach 200 feet above ground?

Answer: 1.58 and 5.92 seconds

d) When does the ball hit the ground?

Answer: After 7.896 seconds

Rational and Radical Functions

15. Find the domain of the following functions.

a) ?(?) =

?

?−3

− 2

Answer: (−∞, 3) ∪ (3, ∞)

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 6 of 12

b) ?(?) = 1 − √3? − 6

Answer: [2, ∞)

16. Given functions ?(?) = √? − 3, find:

a) ?(7) − 2

Answer: 0

b) the domain of the function ? which is define as ?(?) = ?(?) − 2.

Answer: [3, ∞)

17. Simplify the following rational expression.

a)

?

?+2

+

2

?−4

Answer:

?2−2?+4

(?−4)(?+2)

b)

4

?

−

3

?−5

+

5

?+3

Answer:

6(?2−7?−10)

?(?−5)(?+3)

c)

?+2

?+8

−

18

?2+13?+40

Answer:

?−1

?+5

d)

1

3+ℎ

−

1

3

ℎ

Answer: −

1

3(3+ℎ)

18. Solve for ? from the following equation.

a) ? =

?−2

?+3

Answer: ? = −

3?+2

?−1

b) ? = √3? + 10

Answer: ? = 5 (from quadratic, reject ? = −2)

c) ? − √3? − 2 = 4

Answer: ? = 9 (from quadratic, reject ? = 2)

d)

?+2

?+1

+

5

?2+4?+3

− 1 = 0

Answer: ? = −8 (from linear)

e)

?2

?−3

+ 2 =

12−?

?−3

Answer: ? = −6 (from quadratic, reject ? = 3.)

Exponential and Logarithmic Functions

19. Solve for ? to the nearest hundredth from: ln(2? + 1) = 5.

Answer: 73.71

20. The value V of a car measured in dollars, is given by the formula: ? = 40, XXXXXXXXXX)−0.4?

where ? is the age of the car in years. Find the age of the car, to the nearest hundredth of a

year, when the value of the car will be $20,000.

Answer: 8.37 year.

21. Solve to the nearest hundredth: ?2? = 7 . Answer x ≈ 0.97

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 7 of 12

22. Solve to the nearest tenth: 5 ?−3? = 2 . Answer ? ≈ 0.3 .

23. The function ?(?) = 70?3/4 models the number of calories per day, a person needs to

maintain life in terms of that person's weight, ?, in kilograms. (?=person's weight; ?(?) =

calories needed).

a) If a person weighs 80 kilograms, how many calories per day does this person need to

maintain life? Round your answer to the nearest calorie.

Answer: 1872 calories

b) Mark a point on the graph that conveys the information from part (a). (The graph was

given in the Review sheet).

Answer: Mark the point (80, ?(80)) or (80,1872.47)

24. Solve for ? to the nearest hundredth: 32?+1 = 38.

Answer: ? = 1.16

25. Marcel deposits $9,000 in a bank that gives an annual interest rate of 4 .25% per year, which

is compounded quarterly. What will his balance be after 10 years?

Answer: $13,735.49

26. Consider the logarithmic function ?(?) = log7? + log7(? − 6). Find the value of x such that

?(?) = 1.

Answer: ? = 7 (It’s a coincidence that the answer matches the base); reject ? = −1

27. Solve for the exact value of ? from: log27?

2 − log27? = 2/3.

Answer: ? = 9

28. Solve for the exact value of ? from: 41−3? = 1

Answer: ? =

1

3

29. Mike invests $6,700 at 5.1% annual interest compounded monthly. He wants to know how

long he should invest the money for his investment to triple in value. You may use the

formula ? = ?(1 +

?

?

)??. Round to the nearest hundredth.

Answer: approximately 21.59 years

30. Find the domain of the function: ?(?) = ln(5 − 3?). Express your answer in interval notation.

Answer: (−∞,

5

3

)

31. Solve for ? from: log4? + log4(? − 6) = 2 by

a) finding an equivalent equation without logarithms,

Answer: ?(? − 6) = 42

b) and solving this equation.

Answer: ? = 8; reject ? = −2

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 8 of 12

32. Solve for ? from: log(16 − 20?) − log(−?) = 2

Answer: ? = −

1

5

33. The amount ? in grams, of a radioactive material remaining after t days is given by ? =

100?−0.004?. Find the number of days, to the nearest hundredth, when there will be 60 grams

left.

Answer: XXXXXXXXXXdays

Absolute Value Functions

34. Consider the function ?(?) = 3|2? − 5| − 4. Find all values of ? for which ?(?) = 5. Write

your answer in set-builder notation.

Answer: ? = {1,4}

35. Consider the function ?(?) = −2|4? − 3| + 7. Find all values of ? for which ?(?) = −3. Write

your answer in set-builder notation.

Answer: ? = {−

1

2

, 2}

Slopes of parallel/perpendicular lines

36. Write the equation of the line passing through (−2,3) and parallel to the line 3? − 2? = 5.

Express your answer in the slope-intercept form.

Answer: ? =

3

2

? + 6

37. Write the equation of the line that passes through (−6,7) and perpendicular to the line that

has ?-intercept of 2 and ?-intercept of −4. Express your answer in the slope-intercept form.

Answer: ? = −

1

2

? + 4

38. Let ?1 be a line whose equation is 5? + 3? = 6 and ?2 be a line that passes through the points

(−2,0) and (3, −6). Determine if the lines ?1 and ?2 are parallel, perpendicular or neither.

Answer: neither

Linear Inequalities and Interval Notations

39. Solve the following inequalities. Use interval notation to express the solution sets.

a) 1 −

1

3

(2? + 5) ≥ 2 − 6?

Answer: [

1

2

, ∞)

b) −3 ≤

2?−1

3

< 5

Answer: [−4,8)

c) −6 ≤

−2?−4

2

< −3

Answer: (1,4]

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 9 of 12

Radicals, Rational Exponents and Complex Numbers

40. Simplify. Express your results with positive exponents only.

(

?

5

2?−5

27?−

1

2?

)

−

2

3

Answer:

9?4

?2

41. Simplify the following expression and write your answer in radical notation.

a) (

?

−

1

3

√?2

3 )

−3

Answer: ?2?

b) (?−3?

1

2)

2

3

Answer:

√?

3

?2

c) √48x8y

5

Answer: 4?4?2√3?

d) √8?4

3

− 8√?7

3

Answer: (2? − 8?2)√?

3

e) √27?9

3

+ √?6

3

Answer: 3?3 + ?2

f)

√?

1−√?

Answer:

?+√?

1−?

.

42. Evaluate the polynomial ?2 − 3? + 5 at ? = 1 + ?. Express your answer in the form ? + ??.

Answer: 2 − ?

43. Evaluate the expression ?2 + 5? − 3? when ? = −3 + 4?. Express your answer in the form ? +

??.

Answer: −22 − 7?

44. Show whether x = 1+2i is a solution of ?2 − 2? + 5 = 0 .

45. Solve ?2 + 9 = 0 for the complex number z. Answer z = +3i or Z=-3i.

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 10 of 12

Geometric problems

46. Use basic geometry is used to set up a quadratic equation and solve the problem.

a) A circle is inscribed in a square as shown. The area of the square is 16 cm2. Find the

area of the shaded region.

Answer:

16−4?

4

≈ 0.86 cm2

b) A circle is inscribed in a square as shown. The two shown diameters are

perpendicular. The area of the circle is 10 cm2. Find the length of the dashed line

segment.

Answer: √

20

?

≈ 2.52 cm

c) The area between two concentric circles is 25 cm2. The radius of the outer circle is 1

cm more than the radius of the inner circle. Find the length of each radius.

Answer: The radius of the inner circle is

25−?

2?

≈ 3.48 cm. The radius of the outer

circle is

25+?

2?

≈ 4.48 cm.

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 11 of 12

d) A circle is inscribed in a square as shown. The two shown diameters are

perpendicular. The side of the square is 10 cm . Find the area of the shaded region.

Answer: 25 − 25?/4 cm2.

e) A rectangle is shown inscribed in a circle. The rectangle’s diagonals bisect each other

at the circle’s center. The area of the circle is 16 π cm2. The length of ?? is 3. Find the

length of the segment ??.

Answer: ?? = √??2 − ??2 = √4?2 − 9 = √64 − 9 = √55 cm.

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 12 of 12

Answer to Question 10 d) in Quadratic Functions:

Graph the quadratic functions ?(?) = −2?2 − 4? + 5

Without graphing, we know the parabola will open downwards because ? = −2 < 0 (a is the

coefficient of the quadratic term)

a) The coordinates of all intercepts

Answer: shown on graph

b) The coordinates of the vertex

Answer: shown on graph

c) The equation and graph of the axis of symmetry

Answer: shown on graph; ? = −1

d) The domain and range in interval notation

Answer: Domain=(−∞, ∞) Range=(−∞, 7)

e) The coordinates of an additional point on the graph

Answer: shown on graph

f) The maximum or minimum range value

Answer: ? = 7

g) The domain value at which the max or min is reached

Answer: ? = −1

h) ?-interval(s) at which the function is negative

Answer: (−∞,

−2−√14

2

) ∪ (

−2+√14

2

, ∞) ≈ (−∞, −2.871) ∪ (0.871, ∞)

i) ?-interval(s) at which the function is positive

Answer: (

−2−√14

2

,

−2+√14

2

) ≈ (−2.871,0.871)

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 1 of 12

Graphs and Functions

1. Use the graph of the function, ? = ?(?), to answer the question.

a) Estimate ?(1)

Answer: ?(1) ≈ 5 (find a point on the graph that has ?-coordinate = 1).

b) Estimate the solution(s) to ?(?) = 9

Answer: ? ≈ 1.75 (find all points on the graph that have ?-coordinate=9; in this case,

there's just one such point).

c) Estimate the coordinates of all intercepts.

Answer: ?-intercept is approximately(−8.5,0). (All ?-intercepts have ?-coordinate=

0.) There’s only one ?-intercept in this graph; ?-intercept is approximately(0,3). (All ?-

intercepts have ?-coordinate= 0. There’s only one y-intercept in this graph.)

d) Estimate the domain and range of the function

Answer: Domain: (−∞, ∞). (The arrows indicate that the graph extends, without

breaks, indefinitely to the left and indefinitely to the right.) Range: (−∞, ∞). (The

arrows indicate the graph extends, without breaks, indefinitely up and indefinitely

down.)

e) Estimate the minimum or maximum ?-value

Answer: no minimum ?-value and no maximum ?-value. (The arrows indicate that the

graph extends indefinitely up and indefinitely down.)

f) Estimate the ?-value at which the minimum or maximum is achieved?

Answer: no such ?-values because of the answer to part e).

g) Estimate the values of ? for which ? < 0

Answer: (−∞, 8.5) For these ?-values the graph is below the ?-axis so for these ?-

values, ?<0. See how this answer relates to part c)

h) When 1 < ? < 2, is the function increasing or decreasing?

Answer: Increasing. Move to the right within the stated interval of ?-values. The graph

is going upwards for those ?-values.

2. Use the graph of the function, ? = ?(?), to answer the question.

a) Estimate ?(−11)

Answer: ? = ?(−11) ≈ 7

b) Estimate the solution(s) to ?(?) = −4

Answer: ? ≈ −13 and ? ≈ −6

c) Estimate the coordinates of all intercepts.

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 2 of 12

Answer: ?-intercepts: (−14,0), (0,0), (9,0); ?-intercept: (0,0)

d) Estimate the domain and range of the function

Answer: Domain (−∞, ∞) Range [−7.75, ∞)

e) Estimate the minimum or maximum ?-value

Answer: minimum ?-value ≈ −7.75; no maximum ?-value

f) Estimate the x-value at which the minimum or maximum is achieved?

Answer: For minimum: ? ≈ −10 (The ?-coordinate of the point with the minimum ?-

value); For maximum: no such ?-value

g) Estimate the values of ? for which ? < 0

Answer: (−14,0) ∪ (0,9) (the symbol ∪ stands for union or “together with”. The

endpoints are not included because there we have ? = 0 not ? < 0

h) When 4 < ? < 5, is the function increasing or decreasing?

Answer: decreasing

3. Compute

?(4)−?(1)

4−1

for the given function.

a) ?(?) = ?2 − 3? + 1

Answer: 2

b) ?(?) = √?

Answer:

1

3

Linear Functions

4. Suppose that a manufacturer wishes to use a linear function to model how the demand for its

product affects the product’s price. The unit price function has the form ?(?) = ?? + ? where

? is the number of units of the product in demand. Suppose that when 2400 units are

demanded then the unit price is $250 and when 3000 units are demanded then the unit price

is $300.

Number of units ? Price per Unit ?

XXXXXXXXXX

XXXXXXXXXX

a) Determine the equation of the function ?(?) = ?? + ?

b) According to this model, what is the unit price ?(?) if the demand is ? = 3600?

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 3 of 12

Answer: a) ?(?) =

1

12

? + 50 b) $350

5. A linear function ? passes through (−1,5) and (2, −4). Find the function value ?(−2).

Answer: ?(−2) = 8

6. Markos currently has 200 songs in his music collection. Every month, he adds 15 new songs.

Write a formula for the number of songs, ?, in his collection as a function of time, ?, the

number of months. How many songs will he own in a year?

Answer: ?(?) = 15? + 200; 380 songs

7. A town’s population has been growing linearly. In 2004 the population was 6,200. By 2009

the population had grown to 8,100. Assume this trend continues.

a) Write a linear equation to model the population growth.

Answer: ?(?) = 380? + 6200, where ? is the number of years since 2004.

b) Predict the population in 2013.

Answer: 9,620

c) Identify the year in which the population will reach 15,000.

Answer: Sometime around the year 2027

Quadratic Equations (Factoring and Quadratic Formula)

8. Factor completely. 40?4 + 22?3 − 6?2

Answer: 2?2(4? + 3)(5? − 1)

9. Solve the quadratic equations

a) 4?2 + ? = 3

Answer: ? = −1 or ? = −

3

4

b) (? + 1)(? − 3) = 2

Answer: ? = 1 ± √6

c) ?2 + 7? + 12 = 0

Answer: ? = −3 or ? = −4

d) 2?2 + 1 = ?

Answer: ? =

1±?√?

4

e) 2(? XXXXXXXXXX = 0

Answer: ? = −3 ± 2?

f)

1

3

?2 + ? = −

1

2

Answer: ? =

−3±√3

2

g) 4?2 − ? − 3 = 0

Answer: ? = 1 or ? = −

3

4

h) (? −

1

3

)

2

−

4

9

= 0

Answer: ? = 1 or ? = −

1

3

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 4 of 12

Quadratic Functions

10. Graph the following quadratic functions.

a) ?(?) = −?2 − 6? − 5

b) ℎ(?) = −(? − 3)(? + 2)

c) ?(?) = 2?2 − 3? − 2

d) ?(?) = −2?2 − 4? + 5

e) ?(?) = (2? − 1)(? + 3)

f) ?(?) = 2?2 + 8

Before graphing the parabolas, state which open upwards and which open downwards.

For each parabola include

a) The coordinates of all intercepts

b) The coordinates of the vertex

c) The equation and graph of the axis of symmetry

d) The domain and range in interval notation

e) The coordinates of an additional point on the graph

f) The maximum or minimum range value

g) The domain value at which the max or min is reached

h) ?-interval(s) at which the function is negative

i) ?-interval(s) at which the function is positive

Application of Quadratic Equations/Functions

11. The length of a rectangle exceeds twice its width by 3 inches. If the area is 10 square inches,

find the rectangle's dimensions. Round to the nearest tenth of an inch.

Answer: Approximately 1.6 in by 6.2 in.

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 5 of 12

12. The hypotenuse of a right triangle is 6 feet long. One leg is 2 feet shorter than the other. Find

the lengths of the legs. Round to the nearest tenth of a foot.

Answer: Approximately 5.1 ft and 3.1 ft.

13. A retailer estimates that, by charging x dollars each for a particular phone case, she can sell

40 − ? units each week. The function ?(?) = ?(40 − ?), models the weekly revenue, ?(?),

received when the selling price is ?.

a) Interpret (0, ?(0)) and (1, ?(1)) in context.

Answer: When the charge is $0, the revenue is $0; When the charge is $1, the revenue

is $39

b) Find the selling price that will give the maximum revenue, and then find the amount of

the maximum revenue.

Answer: The maximum revenue is $400 when the charge is $20

c) For which selling price(s) will the weekly revenue equal $300?

Answer: $10 and $30

14. A ball is thrown vertically upward, at the rate of 120 ft/sec, from a rooftop with a height of 50

feet. The quadratic function ℎ(?) = −16?2 + 120? + 50 models the height of the ball, from the

ground, at time ?.

a) Interpret (0, ℎ(0)) and (1, ℎ(2)) in context.

Answer: 0 seconds after the ball is thrown the ball is 50 feet from the ground; 1

second after the ball is thrown the ball is 154 feet from the ground.

b) Find how long it will take the ball to reach its maximum height, and then find the

maximum height.

Answer: The maximum height of 275 feet is reached in 3.75 seconds

c) At which time(s) does the ball reach 200 feet above ground?

Answer: 1.58 and 5.92 seconds

d) When does the ball hit the ground?

Answer: After 7.896 seconds

Rational and Radical Functions

15. Find the domain of the following functions.

a) ?(?) =

?

?−3

− 2

Answer: (−∞, 3) ∪ (3, ∞)

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Spring 2021 Final Review Questions

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b) ?(?) = 1 − √3? − 6

Answer: [2, ∞)

16. Given functions ?(?) = √? − 3, find:

a) ?(7) − 2

Answer: 0

b) the domain of the function ? which is define as ?(?) = ?(?) − 2.

Answer: [3, ∞)

17. Simplify the following rational expression.

a)

?

?+2

+

2

?−4

Answer:

?2−2?+4

(?−4)(?+2)

b)

4

?

−

3

?−5

+

5

?+3

Answer:

6(?2−7?−10)

?(?−5)(?+3)

c)

?+2

?+8

−

18

?2+13?+40

Answer:

?−1

?+5

d)

1

3+ℎ

−

1

3

ℎ

Answer: −

1

3(3+ℎ)

18. Solve for ? from the following equation.

a) ? =

?−2

?+3

Answer: ? = −

3?+2

?−1

b) ? = √3? + 10

Answer: ? = 5 (from quadratic, reject ? = −2)

c) ? − √3? − 2 = 4

Answer: ? = 9 (from quadratic, reject ? = 2)

d)

?+2

?+1

+

5

?2+4?+3

− 1 = 0

Answer: ? = −8 (from linear)

e)

?2

?−3

+ 2 =

12−?

?−3

Answer: ? = −6 (from quadratic, reject ? = 3.)

Exponential and Logarithmic Functions

19. Solve for ? to the nearest hundredth from: ln(2? + 1) = 5.

Answer: 73.71

20. The value V of a car measured in dollars, is given by the formula: ? = 40, XXXXXXXXXX)−0.4?

where ? is the age of the car in years. Find the age of the car, to the nearest hundredth of a

year, when the value of the car will be $20,000.

Answer: 8.37 year.

21. Solve to the nearest hundredth: ?2? = 7 . Answer x ≈ 0.97

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22. Solve to the nearest tenth: 5 ?−3? = 2 . Answer ? ≈ 0.3 .

23. The function ?(?) = 70?3/4 models the number of calories per day, a person needs to

maintain life in terms of that person's weight, ?, in kilograms. (?=person's weight; ?(?) =

calories needed).

a) If a person weighs 80 kilograms, how many calories per day does this person need to

maintain life? Round your answer to the nearest calorie.

Answer: 1872 calories

b) Mark a point on the graph that conveys the information from part (a). (The graph was

given in the Review sheet).

Answer: Mark the point (80, ?(80)) or (80,1872.47)

24. Solve for ? to the nearest hundredth: 32?+1 = 38.

Answer: ? = 1.16

25. Marcel deposits $9,000 in a bank that gives an annual interest rate of 4 .25% per year, which

is compounded quarterly. What will his balance be after 10 years?

Answer: $13,735.49

26. Consider the logarithmic function ?(?) = log7? + log7(? − 6). Find the value of x such that

?(?) = 1.

Answer: ? = 7 (It’s a coincidence that the answer matches the base); reject ? = −1

27. Solve for the exact value of ? from: log27?

2 − log27? = 2/3.

Answer: ? = 9

28. Solve for the exact value of ? from: 41−3? = 1

Answer: ? =

1

3

29. Mike invests $6,700 at 5.1% annual interest compounded monthly. He wants to know how

long he should invest the money for his investment to triple in value. You may use the

formula ? = ?(1 +

?

?

)??. Round to the nearest hundredth.

Answer: approximately 21.59 years

30. Find the domain of the function: ?(?) = ln(5 − 3?). Express your answer in interval notation.

Answer: (−∞,

5

3

)

31. Solve for ? from: log4? + log4(? − 6) = 2 by

a) finding an equivalent equation without logarithms,

Answer: ?(? − 6) = 42

b) and solving this equation.

Answer: ? = 8; reject ? = −2

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Spring 2021 Final Review Questions

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32. Solve for ? from: log(16 − 20?) − log(−?) = 2

Answer: ? = −

1

5

33. The amount ? in grams, of a radioactive material remaining after t days is given by ? =

100?−0.004?. Find the number of days, to the nearest hundredth, when there will be 60 grams

left.

Answer: XXXXXXXXXXdays

Absolute Value Functions

34. Consider the function ?(?) = 3|2? − 5| − 4. Find all values of ? for which ?(?) = 5. Write

your answer in set-builder notation.

Answer: ? = {1,4}

35. Consider the function ?(?) = −2|4? − 3| + 7. Find all values of ? for which ?(?) = −3. Write

your answer in set-builder notation.

Answer: ? = {−

1

2

, 2}

Slopes of parallel/perpendicular lines

36. Write the equation of the line passing through (−2,3) and parallel to the line 3? − 2? = 5.

Express your answer in the slope-intercept form.

Answer: ? =

3

2

? + 6

37. Write the equation of the line that passes through (−6,7) and perpendicular to the line that

has ?-intercept of 2 and ?-intercept of −4. Express your answer in the slope-intercept form.

Answer: ? = −

1

2

? + 4

38. Let ?1 be a line whose equation is 5? + 3? = 6 and ?2 be a line that passes through the points

(−2,0) and (3, −6). Determine if the lines ?1 and ?2 are parallel, perpendicular or neither.

Answer: neither

Linear Inequalities and Interval Notations

39. Solve the following inequalities. Use interval notation to express the solution sets.

a) 1 −

1

3

(2? + 5) ≥ 2 − 6?

Answer: [

1

2

, ∞)

b) −3 ≤

2?−1

3

< 5

Answer: [−4,8)

c) −6 ≤

−2?−4

2

< −3

Answer: (1,4]

HSI QCC MA-119

Spring 2021 Final Review Questions

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Radicals, Rational Exponents and Complex Numbers

40. Simplify. Express your results with positive exponents only.

(

?

5

2?−5

27?−

1

2?

)

−

2

3

Answer:

9?4

?2

41. Simplify the following expression and write your answer in radical notation.

a) (

?

−

1

3

√?2

3 )

−3

Answer: ?2?

b) (?−3?

1

2)

2

3

Answer:

√?

3

?2

c) √48x8y

5

Answer: 4?4?2√3?

d) √8?4

3

− 8√?7

3

Answer: (2? − 8?2)√?

3

e) √27?9

3

+ √?6

3

Answer: 3?3 + ?2

f)

√?

1−√?

Answer:

?+√?

1−?

.

42. Evaluate the polynomial ?2 − 3? + 5 at ? = 1 + ?. Express your answer in the form ? + ??.

Answer: 2 − ?

43. Evaluate the expression ?2 + 5? − 3? when ? = −3 + 4?. Express your answer in the form ? +

??.

Answer: −22 − 7?

44. Show whether x = 1+2i is a solution of ?2 − 2? + 5 = 0 .

45. Solve ?2 + 9 = 0 for the complex number z. Answer z = +3i or Z=-3i.

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 10 of 12

Geometric problems

46. Use basic geometry is used to set up a quadratic equation and solve the problem.

a) A circle is inscribed in a square as shown. The area of the square is 16 cm2. Find the

area of the shaded region.

Answer:

16−4?

4

≈ 0.86 cm2

b) A circle is inscribed in a square as shown. The two shown diameters are

perpendicular. The area of the circle is 10 cm2. Find the length of the dashed line

segment.

Answer: √

20

?

≈ 2.52 cm

c) The area between two concentric circles is 25 cm2. The radius of the outer circle is 1

cm more than the radius of the inner circle. Find the length of each radius.

Answer: The radius of the inner circle is

25−?

2?

≈ 3.48 cm. The radius of the outer

circle is

25+?

2?

≈ 4.48 cm.

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 11 of 12

d) A circle is inscribed in a square as shown. The two shown diameters are

perpendicular. The side of the square is 10 cm . Find the area of the shaded region.

Answer: 25 − 25?/4 cm2.

e) A rectangle is shown inscribed in a circle. The rectangle’s diagonals bisect each other

at the circle’s center. The area of the circle is 16 π cm2. The length of ?? is 3. Find the

length of the segment ??.

Answer: ?? = √??2 − ??2 = √4?2 − 9 = √64 − 9 = √55 cm.

HSI QCC MA-119

Spring 2021 Final Review Questions

Page 12 of 12

Answer to Question 10 d) in Quadratic Functions:

Graph the quadratic functions ?(?) = −2?2 − 4? + 5

Without graphing, we know the parabola will open downwards because ? = −2 < 0 (a is the

coefficient of the quadratic term)

a) The coordinates of all intercepts

Answer: shown on graph

b) The coordinates of the vertex

Answer: shown on graph

c) The equation and graph of the axis of symmetry

Answer: shown on graph; ? = −1

d) The domain and range in interval notation

Answer: Domain=(−∞, ∞) Range=(−∞, 7)

e) The coordinates of an additional point on the graph

Answer: shown on graph

f) The maximum or minimum range value

Answer: ? = 7

g) The domain value at which the max or min is reached

Answer: ? = −1

h) ?-interval(s) at which the function is negative

Answer: (−∞,

−2−√14

2

) ∪ (

−2+√14

2

, ∞) ≈ (−∞, −2.871) ∪ (0.871, ∞)

i) ?-interval(s) at which the function is positive

Answer: (

−2−√14

2

,

−2+√14

2

) ≈ (−2.871,0.871)

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