Math III Final Exam Review...3 Evaluate. Round to three decimal places if necessary. 8. log50 9. log 32 64 10. log 5 13 11. ln12 12. log 3 17 13. lnex 2 14. log 6 216 15. x3 log 3

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Math III – Final Exam Review

Unit 1 – Functions & their Graphs

Describe how the graph of 𝑦 = |𝑥| changes to produce each of the following graphs.

1. 𝑦 = −2|𝑥 − 5| 2. 𝑦 =1

3|𝑥 + 1| − 2 3. 𝑦 = |3𝑥 − 9| + 2

Solve the following absolute value equations.

4. |15 − 𝑥| = 12 5. −2|𝑥 + 7| + 5 = 3 6. |4𝑥 − 3| + 1 = 12

Solve the inequality and write the solution in INTERVAL NOTATION.

7. 1023 −x 8. 528 − x 9. 532 +− xx

Graph the following piecewise functions.

10. 𝑓(𝑥) = {4 − 3𝑥 𝑥 ≥ −1|𝑥 + 2| 𝑥 < −1

11. 𝑓(𝑥) = {−2𝑥 − 1 𝑥 ≤ −2(𝑥 + 1)2 − 3 𝑥 > −2

* 12. Find 2f(-2) + f(4) – 3f(6)

𝑓(𝑥) = 𝑓(𝑥) = {−3, 𝑥 < 0

−3𝑥 − 4, −2 < 𝑥 ≤ 4−2𝑥 + 3, 𝑥 > 4

Solve the following systems algebraically.

13. {𝑥2 + 𝑦2 = 9𝑦 = −𝑥 − 3

14. {𝑦 = (𝑥 + 2)2 − 3

𝑦 = 𝑥 + 1

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Unit 2 - Exponential & Logarithmic Functions

**Graph the following and state the domain range, asymptotes, & intercepts. (Non Calc)

1. 𝑦 = 2 ∙ 3−𝑥 2. 𝑦 = 3𝑥+1 − 2 3. 𝑦 = −3 ∙ (1

2)

𝑥+ 1

Asymptote: Asymptote: Asymptote:

Domain: Domain: Domain:

Range: Range: Range:

x-int: x-int: x-int:

y-int: y-int: y-int:

4. 2log)( 4 −= xxf 5. ( )1log)( 2 += xxg

Asymptote: Asymptote:

Domain: Domain:

Range: Range:

x-int: x-int:

y-int: y-int:

6. ( ) 2log)( 3 −−= xxf 7. ( ) 1log)( 3 +−= xxg

Asymptote: Asymptote:

Domain: Domain:

Range: Range:

x-int: x-int:

y-int: y-int:

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Evaluate. Round to three decimal places if necessary.

8. 50log 9. 64log32 10. 13log5

11. 12ln 12. 17log3 13. 2ln +xe

14. 216log6 15. ( )x5log33 16. 51

2 2log

Solve each equation for x.

17. 24 1255 += xx

18.

x

x

=+

32

12 18 19. 304.2 4 =+x

20. 1243 =x 21. 245 4 =+x 22. 5 12314 =−−xe

23. ( ) ( )14log72log +=+ xx 24. 49loglog2 22 =x

25. ( ) ( ) 2log15log2log3log −−=++ xx 26. 3125log =x

27. ( ) 1045ln2 =−−x 28. 22log3log4 22 =−x

29. ( )75log4log +=+ xx 30. 4loglog)13(log 222 =+− xx

31. ( ) ( ) 34log3log 22 =++− xx 32. 3)15(log4 =−x

Write the formula necessary for each of the following and answer the question.

* 33. You deposit $4000 in an account that pays 2.5% annual interest. Find the balance after 5 years if

the interest is compounded annually. Write logarithmic equation but do not solve.

34. You purchase a tractor for $22,500. The value of the tractor decreases by 7% per year. How long will it

take for the value of the tractor to be $14,500?

35. You deposit $1700 in an account that pays 5% annual interest compounded continuously. What

is the balance after 7 years?

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36. Uranium has a half-life of 45 years. How much of a 50-g sample of Uranium will remain after 180 years?

37. If $25,000 is deposited into an account that is compounded continuously at a rate of 4.5% per year. How

long will it take the account to double in value?

* 38. Mary has two job offers out of college. Company #1 offered her $20,000 / year with a guaranteed pay increase of

10% per year. Company #2 offered her $30,000 with a guaranteed pay increase of $2000 per year. How long would she

have to work for Company #1 for her salary to exceed that of Company #2 rounded to the nearest year?

**Find the inverse of the following and state the inverse function’s Domain & Range.

39. 𝑦 = (𝑥 − 2)2 + 3 40. 𝑦 = 5𝑥 − 3 41. 𝑦 = √𝑥 + 4 − 1

42. The table of values represents all points in the function g(x).

What is the value of 𝑔−1(3)?

Unit 3 – Polynomials

1. Use completing the square to put the parabola into vertex form and find the intercepts. 𝑦 = 2𝑥2 + 8𝑥 − 12

2. Simplify: (2𝑥3 − 𝑥2 − 9𝑥 + 9) ÷ (2𝑥 − 3) 3. Simplify: (2𝑥3 + 10𝑥2 + 9𝑥 + 38) ÷ (𝑥 + 5)

Solve 4 & 5 using the quadratic formula

4. 3𝑥2 + 𝑥 = 5 5. −2𝑥 = 7 + 𝑥2

6. Solve. 𝑥4 − 4𝑥2 − 45 = 0

x g(x)

-6 3

-3 9

0 -3

3 -1

5 6

5

7. A projectile is fired upwards from the ground. The

height of the projectile above the ground is shown in

the following table:

a) Find a quadratic model for the data.

b) What is the maximum height?

c) At what time does it reach the maximum height?

d) When does it hit the ground?

* 8. Given: 𝑓(𝑥) = |𝑥 − 5| + 3 and the table:

Describe the intervals over which the f(x) and g(x) are increasing, and

decreasing. State the min or max for each function.

* 9. Find the zeros, y-intercept and minimum: 𝑓(𝑥) = 4𝑥2 + 7𝑥 − 15

* 10. For polynomial 𝑓(𝑥) = 2𝑥3 − 2𝑥2 − 4, f(2) = 4. What does this mean?

* 11. The expression (x + 3) is a factor of 2𝑥2 + 𝑘𝑥 − 15. What is the value of k?

12. Given: 𝑓(𝑥) = 2𝑥2(𝑥 + 3)2(𝑥 − 2), find the following…

a. Degree ______________

b. Zeros_______________

c. Y-intercept: ______________

d. Describe the end behavior. Use correct notation.

e. Sketch the graph with the correct x & approximate y-intercepts

Time (seconds) 0 0.5 1 1.5 2 2.5

Height (feet) 0 20.5 31.36 36.25 30.41 28.23

x g(x)

-3 4

-2 5

-1 6

0 5

1 4

2 3

3 2

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* 13. Find the turning points of the function : 𝑓(𝑥) = 2𝑥3 − 3𝑥2 − 36𝑥 − 2

* 14. For (6𝑥2 − 13𝑥 + 𝑘) ÷ (3𝑥 + 1), find the value of k so that the remainder is 2

* 15. Using the remainder theorem, evaluate ℎ(𝑥) = 3𝑥3 − 4𝑥2 − 4𝑥 + 5 at x = -2

* 16. Describe the number and types of roots of the equation 𝑥3 + 9𝑥 = 0.

* 17. Given the polynomial and one of its factors, find the remaining factors: 𝑥3 + 2𝑥2 + 16𝑥 + 32; (𝑥 + 2)

* 18. Write the polynomial as a product of linear factors. 𝑓(𝑥) = 𝑥3 − 2𝑥2 − 9𝑥 + 18

* 19. Find all the zeros of the function: 𝑓(𝑥) = 5(3𝑥 + 1)(−2𝑥 − 3)(4𝑥 − 2)

* 20. State the possible number of imaginary zeros of 𝑓(𝑥) = 2𝑥4 − 2𝑥3 + 12𝑥2 − 8𝑥 − 12

* 21. Find all the zeros of the function. 𝑓(𝑥) = 𝑥4 + 3𝑥3 − 9𝑥2 − 15𝑥 + 20

* 22. One zero of the function 𝑥3 + 2𝑥2 − 23𝑥 − 60 is −4. Find the other zeros.

* 23. The area of a rectangle is (𝑥3 + 2𝑥2 − 𝑥 − 2) and the width is (𝑥 + 1). What is the length?

* 24. If 𝑓(𝑥) = −𝑥2 + 36 and 𝑔(𝑥) = −𝑥 + 5, find the solution(s) of: 𝑓(𝑥) = 2𝑔(𝑥)

* 25. Write the simplified polynomial for the given graph?

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Unit 4 – Rational Functions

* 1. Describe the translation of 𝑓(𝑥) =7

𝑥 to 𝑔(𝑥) =

−7

𝑥+3+ 5

2. Simplify: 3𝑥2𝑦

4𝑥𝑦2 °2𝑥3𝑦2

9𝑥2𝑦3

3. Simplify: 𝑥+1

𝑥2−16÷

3𝑥+3

𝑥+4

* 4. Solve: 2

𝑥−3+ 4 =

2

𝑥−3

* 5. Solve: 𝑥+2

𝑥+5+

𝑥−1

𝑥2−25= 1

6. The sum of a number and its reciprocal is 5

2. What is the number?

7. If Tom and Jerry can mow a lawn in 3 hours working together. If Tom mows the lawn in 4 hours alone, how long can

Jerry mow the same lawn?

* 8. Pump R can fill a tank in 8 hours and Pump P can empty it in 12 hours. If both pumps are in operation, how long will it take to fill the tank?

Unit 5 – Geometry

1. Use the diagram to classify each pair of angles as one of the following: alternate interior angles, alternate exterior angles,

same side interiors angles, corresponding angles, or vertical angles

a. ∠1 & ∠4 _____________

b. ∠2 & ∠6 _____________

c. ∠4 & ∠6 _____________

1 2

3 4

5 6

7 8

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For # 2 – 3, RSTU is a rectangle. 2. If 𝑈𝑍 = 𝑥 + 21 and 𝑍𝑇 = 3𝑥 − 15, find 𝑈𝑆.

3. If 𝑚∠𝑆𝑈𝑇 = 3𝑥 + 6 and 𝑚∠𝑅𝑈𝑆 = 5𝑥 − 4, find m∠RZS.

4. In rhombus ABCD, 𝑚∠𝐵𝐴𝐶 = 4𝑥 + 9, 5. Determine whether ABCD is a parallelogram, rectangle,

𝑚∠𝐷𝐶𝐴 = 6𝑥 − 19. Find 𝑚∠𝐴𝐷𝐶. rhombus, or square. Explain and show your work.

A(-3,3) B(2,5) C(4,0) D(-1,-2)

* 6. Old McDonald has a farm with a cylindrical silo to hold his grains. The silo is 20 feet tall,

and has a diameter of 12 feet. Approximately what volume of grain can be held in the silo if it

is filled to the top of the dome?

* 7. The Peachy Pitchers company sells a special pitcher that holds an ice tube in the center to keep the beverages cold.

This is good for the Great Escape Café because most of their seating is outside. If their pitcher is 15 cm in diameter and

40 cm tall, and the ice tube in the center is 8 cm in diameter and 35 cm tall. How much liquid does this pitcher hold with

the ice tube inserted?

See link on canvas page to: Why did the plane cross the figure.

See link on canvas page to: Volume of 2-D rotations

Z

R

U

S

T

A B

C D

F

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Unit 6 – Circles & Triangle Centers

1. Find the center and radius for the following circle: 𝑥2 + 𝑦2 + 4𝑥 − 6𝑦 − 36 = 0

* 2. In ∆𝐴𝐵𝐶, D is the midpoint of 𝐴𝐵̅̅ ̅̅ and the distance from C to D is 120 ft.

If P is the centroid of the triangle, what is the distance from P to C?

In ∆𝑷𝑹𝑺, 𝑷𝑻̅̅ ̅̅ is an altitude and 𝑷𝑿̅̅ ̅̅ is a median.

3. Find RS if RX = x + 7 and SX = 3x – 11

4. Find RT if RT = x – 6 and 𝑚∠𝑃𝑇𝑅 = 8𝑥 − 6

In ∆𝑫𝑬𝑭, 𝑮𝑰̅̅̅̅ is a perpendicular bisector.

5. Find x if EH = 16 and FH = 7x – 5 6. Find y if EG = 3.2y – 1 and FG = 2y + 5

* 7. Find x * 8. If P is the incenter of triangle ABC,

find CP if XP = 9 and BX = 12.

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9. AD 10. DC 11. 𝑚𝐴�̂�

12. In circle Z, 𝑃𝑍 = 𝑍𝑄, 𝑋𝑌 = 3𝑥 − 4, 𝑎𝑛𝑑 13. Find the measure of each numbered angle.

𝑆𝑇 = −6𝑥 + 23. Find SQ.

Assume that segments that appear to be tangent are tangent.

14. Find x. 15. Find AB.

16. Find the perimeter of ∆𝐴𝐵𝐶. 17. Find x.

A B

C

x°

107°

11

Find x.

18. 19. 20.

Unit 7 – Trig

Rewrite the following angle measures in radians. Rewrite the following angle measures in degrees.

1. 300° 2. 225° 3. 7𝜋

4 4. 9

5. Find the values for 𝑥 and 𝑦. Then find the perimeter of the figure.

x = __________

y = _________

Perimeter = ___________________

6. Find the arc length of an arc subtended by an angle whose measure is 3 radians with a radius of 2.

Use circle C for #7 – 8

7. Find the length of minor arc AB in the figure to the right.

The radius is 8 and 𝑚∠𝐴𝐶𝐵 = 45°.

8. Find the area of the shaded (grid) sector of circle C to the right.

The radius is 8 and 𝑚∠𝐴𝐶𝐵 = 45°.

7

x

y

12

* 9. If the center of a ferris wheel is located at (0, 0) and has a radius of 20

meters, what would the:

a. Horizontal distance from the center be if someone got on the ride and

rotated 135 degrees counterclockwise?

b. Vertical distance at 135 degrees counterclockwise

c. Horizontal distance from the center at 60 degrees

d. Vertical distance from the center at 60 degrees

e. Horizontal distance from the center at 120 degrees

f. Vertical distance from the center at 120 degrees

* 10. A pizza is divided into 12 equal slices. If the diameter of the pizza is 16 inches, what is the approximate

arc length of one slice of pizza? (leave in terms of 𝜋). What is the arc length of 3 slices of pizza?

* 11. What is the degree measure of the angle that intercepts the arc of 5𝜋

4?

Unit 8 – Stats

1. Find the sample size required to achieve the given margin of error. Round your answer to the nearest whole number.

A) ± 6.7% B) ± 4% C) ± 3.2% D) ± 2.5%

2. Find the margin of error for a survey that has the given sample size. Round your answer to the nearest tenth of a

percent.

A) 1400 B) 2000 C) 750 D) 200

3. In a survey of 1250 people, 24% said they eat an apple a day.

a) What is the margin of error for this survey?

b) Give an interval that is likely to contain the true proportion of all people who eat an apple a day.