MAT115/117 Final Exam Review

Question 1

Question 2

Question 3

Solve the following inequalities for x. Write your answer in interval notation.

(a) −6x + 7(4x − 24) < (x − 6) − 36

(b) 5.7x − 1.7(x − 7) ≥ 39.9

Solve for . Write your answers, separated by a comma.

(a)

(b)

x

|x + 12| = 15

15 − |x − 14| = − 5

(c) ∣∣8− ∣

∣ = 11x

1

Lumen OHM https://ohm.lumenlearning.com/assess2/?cid=52202&aid=3793629#/print

1 of 11 7/27/2021

Solve for and express your result using interval notations.

(a)

(b)

(c)

(d)

x

|x + 8| − 9 < 10

|1 − x| ≤ 5

|8x − 48| − 16 ≥ 104

3 −∣∣∣

∣∣∣

≤ − 790 + 6x

6

LaGuardia Community College

Mathematics, Engineering, and Computer Science

Question 4

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Find the equation of the line satisfying the speci�ed conditions.

(a) slope of 8 and passing through (1, − 9).

(b) passing through and .

(c) slope of 0 and passing through .

(d) -intercept of and y-intercept of 5.

(e) passing through (4, − 1) with unde�ned slope.

and parallel to the line whose equation is .

Write your answers in slope-intercept form when possible.

( − 6, 3) (9, − 42)

( − 10, − 6)

x −6

(f) through the point ( − 8, − 9) 5x − y = 2

Question 5

Let the quadratic function f be de�ned by f(x) = − 8x − x2 Compute the following.

(a)

(b)

(c) For what values of x is ?

f(2) =

f(0) =

f(x) = 16

Question 6

, .

(b)

f(x + h) − f(x)

hh ≠ 0Compute the di�erence quotient

(a) f(x) = 10 + 3x

f(x) = 2x2 + 6x + 5

Question 7

Question 8

Solve the following systems of linear equations for and . Write your answer as anordered pair.

(a)

(b)

x y

{ −x − 2y = 13

2x − y = −11

{ −6x + 6y = 3

y = −x + 6

The price of a widget was $566. Over six years, the price dropped to $200. Assume a lineartrend.

(a) Find a linear model for this situation, where is the price at time .

(b) Use this model to predict the price in 8 years.

P = ax + b P x

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Question 9 1 pt 1

A jar containing 23 coins (quarters and nickels) has a total value of $3.55. How many ofeach type of coin are there?

Question 10

Cesar invested a total of $41,000 in two accounts. The �rst account earned 7% after one year. However, the second account su�ered a 5% loss in the same time. At the end of one year, the total amount of money gained was $1,670.

How much was invested in each account?

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Question 11

Simplify the following expressions:

(b)

(a) −3( − 4x2 − 3x − 4) − ( − x2 + 3x − 2)

(2x + 3)2

Question 12

Find the quotient and remainder (if any)

(a) ( − 4x3 + 14x2 + 6x + 8) ÷ (x − 4)

(b) ( − 4x3 − 10x2 + 31x + 10) ÷ (x + 4)

Question 13 1 pt 1

Factor completely.

(a)

(b)

(c)

x2 + 4x − 5

49x2 − 98x + 45

8x2 − 200

Question 14 1 pt 1

Solve the quadratic equation by factoring.

Separate your answers by a comma.

27x2 + 54x − 21 = 0

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Question 15

Solve the quadratic equation by completing the square:

−2x2 − 6x − 4 = − 2

Question 16

Solve equation by using the quadratic formula. List the solutions, separated by commas.

4z2 + 3z − 2 = 0

Question 17

Describe the domain for each of the following functions. Write your answer in intervalnotation.

(a)

(b)

(c)

(d)

f(x) =√x − 6x − 15

f(x) = 2x3 − 8x2 − 7x − 9

f(x) =x + 5

x2 − 2x − 35

f(x) = −1

x − 10√x + 1x − 8

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Question 18

(f) Graph h(x)

Consider the function h(x) = x2 − 2x − 8 

(a) What is the vertex of h ?

(b) The x -intercept(s) of h  is/are

(c) The y -intercept of h  is

(d) The domain of h  is

(e) The range of h  is

Question 19

f(x) = ax2 + bx + c whose vertex is (1, 3) and whose y-intercept is (0, 6).Find a quadratic function

Question 20

An object is thrown upward at a speed of 123 feet per second by a machine from a heightof 3 feet o� the ground. The height of the object after seconds can be found using the

, where v0 is the initial velocity and s0 is the initialheight.  Give all numerical answers to 2 decimal places.

(a) When will the object reach its maximum height?

(b) What is its maximum height?

(c) When will the object reach the ground?

t

equation s(t) = − 16t2 + v0t + s0

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Question 21

Conner wants to build a rectangular pen for his animals. One side of the pen will beagainst the barn; the other three sides will be enclosed with wire fencing. If Conner has700 feet of fencing, what dimensions would maximize the area of the pen?

a) Let w be the length of the pen perpendicular to the barn. Write an equation to modelthe area of the pen in terms of

b) What width w would maximize the area?

w

Question 22

Perform the operation and simplify the rational expression.

(a)

(b)

x2 − 25x x − 4

+ =x2 − 5x

÷ =2x2 − 5x − 122x2 − 7x − 4

2x2 + 11x + 12x2 + 2x − 8

Question 23

Solve for x and check for extraneous roots.

(a)

(b)

= + 2x

x + 554

x2 − 25

+ = − 42

x + 9−4x

x − 9

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Question 24

Solve for x and check for extraneous roots.

(a) √x − 9 = 6

(b) √x − 6 = x − 6

(c) √x + 12 = x

(d) √−3x − 11 = − 7 + √4x + 101

Question 25

(b)

(c) .

Find the inverse for each of the following one-to-one functions.

(a) f(x) = − 2x + 4

f(x) =−7x + 82x − 5

f(x) = √x − 5, x ≥ 5

Question 26

f(x) = Cax whose graph goes through the points Find the exponential functionand (3, 24).

(0, 3)

Question 27

Evaluate (without use of a calculator).

(a) log4 2 + log4 128

(b) log4 32 - log4 2

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Question 28

Solve for x without the use of a calculator.

(a) 8x = 16

(c) 3(x^2) = 9

(d) 7√x - 6=49

(b) √x =

Question 29

Solve the equation using logarithm: 77 x - 9 = 59 x - 4

Question 30

Solve for `x` and check your answer.

(a) log5(x-9) = 3

(b) logx(-2 x +35)=2

(c) log2 x + log2(x+2) = 3

(d) log5 x - log5(x + 9) = 3

Question 31

If $35000 dollars is invested at an interest rate of 6 percent per year, �nd the value of theinvestment at the end of 5 years for the following compounding methods, to the nearestcent. Assume no subsequent deposit or withdrawal.

(a) Annual: $

(b) Semiannual: $

(c) Quarterly: $

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Question 32

Question 33

A 500-foot weather tower used to measure wind speed has a guy wire attached to it 175feet above the ground. The angle between the wire and the vertical tower is `17^circ`.Approximate the length of the guy wire (to the nearest foot).

A child is holding the end of a 29 meter string attached to a kit. The string has an angle of elevation of 60°. Determine the elevation of the kite. Give your answer rounded to the nearest whole number.

Question 34

Find the exact value of the following without use of a calculator.

(a) cos(-5π/6)

(b) sin(π/3)

(c) tan(π/4)

Question 35

Given the trigonometric function de�ned by f(x) = sin(4x):

(a) What is the amplitude of the function?

(b) What is the period of the function?

(c) Sketch one period of the function.

Question 36

Sketch y = 2sin(12x) over one period.

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Question 37

Given that c=11 and α = 41°, �nd the values of a and b to the nearest tenths.

Note: The picture isn't drawn to scale.

Question 38

Find the area of the rectangle below, given that z=2 and α = 30°.

Question 39

If A is an acute angle and tan A = (√7 - 2)/(√7 + 2), �nd cos A.

1a. (−∞, 6) 13a. (𝑥𝑥 − 1)(𝑥𝑥 + 5) 1b. [7,∞) 13b. (7𝑥𝑥 − 5)(7𝑥𝑥 − 9) 2a. 𝑥𝑥 = −27, 3 13c. 8(𝑥𝑥 + 5)(𝑥𝑥 − 5) 2b. 𝑥𝑥 = −6, 34 14. 𝑥𝑥 = −7

3, 13

2c. 𝑥𝑥 = −3, 19 15. 𝑥𝑥 = −3±√52

3a. (−27,11) 16. 𝑧𝑧 = −3±√418

3b. [−4,6] 17a. [6,15) ∪ (15,∞)3c. (−∞, 9] ∪ [21,∞) 17b. (−∞,∞)3d. (−∞,−25] ∪ [−5,∞) 17c. (−∞,−5) ∪ (−5,7) ∪ (7,∞)4a. 𝑦𝑦 = 8𝑥𝑥 − 17 17d. [−1,8) ∪ (8,10) ∪ (10,∞)4b. 𝑦𝑦 = −3𝑥𝑥 − 15 18a. (1,−9)4c. 𝑦𝑦 = −6 18b. (4,0), (−2,0)

4d. 𝑦𝑦 = 56𝑥𝑥 + 5 18c. (0,−8)

4e. 𝑥𝑥 = 4 18d. (−∞,∞) 4f. 𝑦𝑦 = 5𝑥𝑥 + 31 18e. [−9,∞) 5a. 𝑓𝑓(2) = −20 18f. see image below 5b. 𝑓𝑓(0) = 0 19. 𝑓𝑓(𝑥𝑥) = 3𝑥𝑥2 − 6𝑥𝑥 + 65c. 𝑥𝑥 = −4 20a. 3.84 seconds6a. 3 20b. 239.39 feet6b. 4𝑥𝑥 + 2ℎ + 6 20c. 7.71 seconds7a. (−7,−3) 21a. 𝐴𝐴 = 2𝑤𝑤2 + 700𝑤𝑤

7b. �114

, 134� 21b. 𝑤𝑤 = 175 feet

8a. 𝑃𝑃 = −61𝑥𝑥 + 566 22a. 2𝑥𝑥2+𝑥𝑥−20𝑥𝑥3−25𝑥𝑥

8b. $78 22b. 𝑥𝑥−22𝑥𝑥+1

9. 11 nickels, 12 quarters 23a. 𝑥𝑥 = −4,−1 10. $31,000 at 7%, $10,000 at 5% 23b. 𝑥𝑥 = −171

17

11a.13𝑥𝑥2 + 6𝑥𝑥 + 14 24a. 𝑥𝑥 = 225 11b. 4𝑥𝑥2 + 12𝑥𝑥 + 9 24b. 𝑥𝑥 = 6,7 12a. Quotient: −4𝑥𝑥2 − 2𝑥𝑥 − 2, Remainder: 0 24c. 𝑥𝑥 = 4 12b. Quotient: −4𝑥𝑥2 + 6𝑥𝑥 + 7, Remainder: −18 24d. 𝑥𝑥 = −5

Answers

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25b. 𝑓𝑓−1(𝑥𝑥) = 5𝑥𝑥+82𝑥𝑥+7

31b. $47,037.07

25c. 𝑓𝑓−1(𝑥𝑥) = 𝑥𝑥2 + 5, 𝑥𝑥 ≥ 0 31c. $47,139.93

26. 𝑓𝑓(𝑥𝑥) = 3 ∙ 2𝑥𝑥 32. 183 feet

27a. 4 33. 25 meters

27b. 2 34a. −√32

28a. 𝑥𝑥 = 43 34b. √3

2

28b. 𝑥𝑥 = 4 34c. 1

28c. 𝑥𝑥 = ±√2 35a. 1

28d. 𝑥𝑥 = 64 35b. 𝜋𝜋2

29. 𝑥𝑥 = 9−4 log7 57−9 log7 5

35c. see image below

30a. 𝑥𝑥 = 134 36. see image below30b. 𝑥𝑥 = 5 37. 𝑎𝑎 = 8.3,𝑏𝑏 = 7.230c. 𝑥𝑥 = 2 38. √3 square units30d. no solution 39. 2√22+√154

22

18f. 35c.

36.

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225a. 𝑓𝑓−1(𝑥𝑥) = −𝑥𝑥+4 31a. $46,837.90