math iii final exam review...3 evaluate. round to three decimal places if necessary. 8. log50 9. log...
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Math III โ Final Exam Review
Unit 1 โ Functions & their Graphs
Describe how the graph of ๐ฆ = |๐ฅ| changes to produce each of the following graphs.
1. ๐ฆ = โ2|๐ฅ โ 5| 2. ๐ฆ =1
3|๐ฅ + 1| โ 2 3. ๐ฆ = |3๐ฅ โ 9| + 2
Solve the following absolute value equations.
4. |15 โ ๐ฅ| = 12 5. โ2|๐ฅ + 7| + 5 = 3 6. |4๐ฅ โ 3| + 1 = 12
Solve the inequality and write the solution in INTERVAL NOTATION.
7. 1023 โx 8. 528 โ x 9. 532 +โ xx
Graph the following piecewise functions.
10. ๐(๐ฅ) = {4 โ 3๐ฅ ๐ฅ โฅ โ1|๐ฅ + 2| ๐ฅ < โ1
11. ๐(๐ฅ) = {โ2๐ฅ โ 1 ๐ฅ โค โ2(๐ฅ + 1)2 โ 3 ๐ฅ > โ2
* 12. Find 2f(-2) + f(4) โ 3f(6)
๐(๐ฅ) = ๐(๐ฅ) = {โ3, ๐ฅ < 0
โ3๐ฅ โ 4, โ2 < ๐ฅ โค 4โ2๐ฅ + 3, ๐ฅ > 4
Solve the following systems algebraically.
13. {๐ฅ2 + ๐ฆ2 = 9๐ฆ = โ๐ฅ โ 3
14. {๐ฆ = (๐ฅ + 2)2 โ 3
๐ฆ = ๐ฅ + 1
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Unit 2 - Exponential & Logarithmic Functions
**Graph the following and state the domain range, asymptotes, & intercepts. (Non Calc)
1. ๐ฆ = 2 โ 3โ๐ฅ 2. ๐ฆ = 3๐ฅ+1 โ 2 3. ๐ฆ = โ3 โ (1
2)
๐ฅ+ 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 โ= xxf 5. ( )1log)( 2 += xxg
Asymptote: Asymptote:
Domain: Domain:
Range: Range:
x-int: x-int:
y-int: y-int:
6. ( ) 2log)( 3 โโ= xxf 7. ( ) 1log)( 3 +โ= xxg
Asymptote: Asymptote:
Domain: Domain:
Range: Range:
x-int: x-int:
y-int: y-int:
3
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 =โโxe
23. ( ) ( )14log72log +=+ xx 24. 49loglog2 22 =x
25. ( ) ( ) 2log15log2log3log โโ=++ xx 26. 3125log =x
27. ( ) 1045ln2 =โโx 28. 22log3log4 22 =โx
29. ( )75log4log +=+ xx 30. 4loglog)13(log 222 =+โ xx
31. ( ) ( ) 34log3log 22 =++โ xx 32. 3)15(log4 =โx
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 functionโs Domain & Range.
39. ๐ฆ = (๐ฅ โ 2)2 + 3 40. ๐ฆ = 5๐ฅ โ 3 41. ๐ฆ = โ๐ฅ + 4 โ 1
42. The table of values represents all points in the function g(x).
What is the value of ๐โ1(3)?
Unit 3 โ Polynomials
1. Use completing the square to put the parabola into vertex form and find the intercepts. ๐ฆ = 2๐ฅ2 + 8๐ฅ โ 12
2. Simplify: (2๐ฅ3 โ ๐ฅ2 โ 9๐ฅ + 9) รท (2๐ฅ โ 3) 3. Simplify: (2๐ฅ3 + 10๐ฅ2 + 9๐ฅ + 38) รท (๐ฅ + 5)
Solve 4 & 5 using the quadratic formula
4. 3๐ฅ2 + ๐ฅ = 5 5. โ2๐ฅ = 7 + ๐ฅ2
6. Solve. ๐ฅ4 โ 4๐ฅ2 โ 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: ๐(๐ฅ) = |๐ฅ โ 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: ๐(๐ฅ) = 4๐ฅ2 + 7๐ฅ โ 15
* 10. For polynomial ๐(๐ฅ) = 2๐ฅ3 โ 2๐ฅ2 โ 4, f(2) = 4. What does this mean?
* 11. The expression (x + 3) is a factor of 2๐ฅ2 + ๐๐ฅ โ 15. What is the value of k?
12. Given: ๐(๐ฅ) = 2๐ฅ2(๐ฅ + 3)2(๐ฅ โ 2), find the followingโฆ
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 : ๐(๐ฅ) = 2๐ฅ3 โ 3๐ฅ2 โ 36๐ฅ โ 2
* 14. For (6๐ฅ2 โ 13๐ฅ + ๐) รท (3๐ฅ + 1), find the value of k so that the remainder is 2
* 15. Using the remainder theorem, evaluate โ(๐ฅ) = 3๐ฅ3 โ 4๐ฅ2 โ 4๐ฅ + 5 at x = -2
* 16. Describe the number and types of roots of the equation ๐ฅ3 + 9๐ฅ = 0.
* 17. Given the polynomial and one of its factors, find the remaining factors: ๐ฅ3 + 2๐ฅ2 + 16๐ฅ + 32; (๐ฅ + 2)
* 18. Write the polynomial as a product of linear factors. ๐(๐ฅ) = ๐ฅ3 โ 2๐ฅ2 โ 9๐ฅ + 18
* 19. Find all the zeros of the function: ๐(๐ฅ) = 5(3๐ฅ + 1)(โ2๐ฅ โ 3)(4๐ฅ โ 2)
* 20. State the possible number of imaginary zeros of ๐(๐ฅ) = 2๐ฅ4 โ 2๐ฅ3 + 12๐ฅ2 โ 8๐ฅ โ 12
* 21. Find all the zeros of the function. ๐(๐ฅ) = ๐ฅ4 + 3๐ฅ3 โ 9๐ฅ2 โ 15๐ฅ + 20
* 22. One zero of the function ๐ฅ3 + 2๐ฅ2 โ 23๐ฅ โ 60 is โ4. Find the other zeros.
* 23. The area of a rectangle is (๐ฅ3 + 2๐ฅ2 โ ๐ฅ โ 2) and the width is (๐ฅ + 1). What is the length?
* 24. If ๐(๐ฅ) = โ๐ฅ2 + 36 and ๐(๐ฅ) = โ๐ฅ + 5, find the solution(s) of: ๐(๐ฅ) = 2๐(๐ฅ)
* 25. Write the simplified polynomial for the given graph?
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Unit 4 โ Rational Functions
* 1. Describe the translation of ๐(๐ฅ) =7
๐ฅ to ๐(๐ฅ) =
โ7
๐ฅ+3+ 5
2. Simplify: 3๐ฅ2๐ฆ
4๐ฅ๐ฆ2 ยฐ2๐ฅ3๐ฆ2
9๐ฅ2๐ฆ3
3. Simplify: ๐ฅ+1
๐ฅ2โ16รท
3๐ฅ+3
๐ฅ+4
* 4. Solve: 2
๐ฅโ3+ 4 =
2
๐ฅโ3
* 5. Solve: ๐ฅ+2
๐ฅ+5+
๐ฅโ1
๐ฅ2โ25= 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 โ 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. โ 1 & โ 4 _____________
b. โ 2 & โ 6 _____________
c. โ 4 & โ 6 _____________
1 2
3 4
5 6
7 8
8
For # 2 โ 3, RSTU is a rectangle. 2. If ๐๐ = ๐ฅ + 21 and ๐๐ = 3๐ฅ โ 15, find ๐๐.
3. If ๐โ ๐๐๐ = 3๐ฅ + 6 and ๐โ ๐ ๐๐ = 5๐ฅ โ 4, find mโ RZS.
4. In rhombus ABCD, ๐โ ๐ต๐ด๐ถ = 4๐ฅ + 9, 5. Determine whether ABCD is a parallelogram, rectangle,
๐โ ๐ท๐ถ๐ด = 6๐ฅ โ 19. Find ๐โ ๐ด๐ท๐ถ. 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 Cafรฉ 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 โ Circles & Triangle Centers
1. Find the center and radius for the following circle: ๐ฅ2 + ๐ฆ2 + 4๐ฅ โ 6๐ฆ โ 36 = 0
* 2. In โ๐ด๐ต๐ถ, D is the midpoint of ๐ด๐ตฬ ฬ ฬ ฬ 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 โ๐ท๐น๐บ, ๐ท๐ปฬ ฬ ฬ ฬ is an altitude and ๐ท๐ฟฬ ฬ ฬ ฬ is a median.
3. Find RS if RX = x + 7 and SX = 3x โ 11
4. Find RT if RT = x โ 6 and ๐โ ๐๐๐ = 8๐ฅ โ 6
In โ๐ซ๐ฌ๐ญ, ๐ฎ๐ฐฬ ฬ ฬ ฬ is a perpendicular bisector.
5. Find x if EH = 16 and FH = 7x โ 5 6. Find y if EG = 3.2y โ 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. ๐๐ด๏ฟฝฬ๏ฟฝ
12. In circle Z, ๐๐ = ๐๐, ๐๐ = 3๐ฅ โ 4, ๐๐๐ 13. Find the measure of each numbered angle.
๐๐ = โ6๐ฅ + 23. Find SQ.
Assume that segments that appear to be tangent are tangent.
14. Find x. 15. Find AB.
16. Find the perimeter of โ๐ด๐ต๐ถ. 17. Find x.
A B
C
xยฐ
107ยฐ
11
Find x.
18. 19. 20.
Unit 7 โ Trig
Rewrite the following angle measures in radians. Rewrite the following angle measures in degrees.
1. 300ยฐ 2. 225ยฐ 3. 7๐
4 4. 9
5. Find the values for ๐ฅ and ๐ฆ. 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 โ 8
7. Find the length of minor arc AB in the figure to the right.
The radius is 8 and ๐โ ๐ด๐ถ๐ต = 45ยฐ.
8. Find the area of the shaded (grid) sector of circle C to the right.
The radius is 8 and ๐โ ๐ด๐ถ๐ต = 45ยฐ.
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 ๐). What is the arc length of 3 slices of pizza?
* 11. What is the degree measure of the angle that intercepts the arc of 5๐
4?
Unit 8 โ Stats
1. Find the sample size required to achieve the given margin of error. Round your answer to the nearest whole number.
A) ยฑ 6.7% B) ยฑ 4% C) ยฑ 3.2% D) ยฑ 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.