Algebra 2
Semester 2 Final REVIEW
Name ____________________________ Per ____
Clearly identify your final solutions by boxing your answers. Show your work on every problem!!
Chapter 6 – Radical Functions
For each function, find the domain, range and transformations from its parent function. Then algebraically find its
inverse.
1. 𝑔(𝑥) = √𝑥 + 5 − 2
3
2. ℎ(𝑥) = √𝑥 − 6 − 1
Domain:
Domain:
Range:
Range:
Transformations:
Transformations:
Inverse:
Inverse:
Simplify each expression. Rationalize all denominators, when applicable.
4
3. √256𝑠 7 𝑧 12
5.
√3
3
4. √625𝑢5 𝑣 8
6.
√2
√6𝑥
√7
1
7. (9𝑥 3 𝑦 4 )2
8. Solve the equation. Check for extraneous
solutions.
√4𝑥 + 3 + 2 = 5
9. Solve the equation. Check for extraneous
solutions.
√𝑥 − 5 = √2𝑥 + 3
Chapter 7 – Logarithms & Exponential
Graph each logarithmic function. List the
transformations.
10. 𝑦 = log 3 𝑥 and 𝑦 = log 3 𝑥 − 5
Graph each logarithmic function. List the
transformations.
11. 𝑦 = log 2 𝑥 and 𝑦 = log 2 (𝑥 + 3)
Solve the equation.
Solve the equation.
12. log3 (x + 1) = 4
13. 4 + 5x = 29
14. Evaluate log4 256
15. Evaluate log 27 9
1
16. Evaluate log 2 32
17. The population of fish in a lake is decreasing. There are 18. Radium-226 has a half-life of 1660 years. An initial
currently 24,000 fish in the lake. The population is
decreasing by 6% each year. In how many years will
1
there be of the current population of fish in the lake?
4
amount of Radium-226 has a mass of 560 kg. Write an
exponential function that models the decay of this
material. In how many years will there only be 25kg of
Radium-226 remaining?
Chapter 8 – Rational Equations
3
19. Write an equation for the translation of 𝑦 = 𝑥 with
the a vertical asymptote of x = -4 and a horizontal
asymptote of y = -3
20. For the rational function, identify any points of
discontinuity.
𝑦=
4(𝑥 − 2)(𝑥 + 8)
𝑥 2 − 9𝑥 + 14
Hole:
Vertical Asymptote:
Horizontal Asymptote:
Domain:
Range:
21. Simplify the rational expression. State any restrictions
on the variable.
10
5
1
=
−
𝑥+8 𝑥−9 𝑥−9
22. Simplify the rational expression. State any restrictions
on the variable.
3
𝑥+2
=
𝑥+4
5
Restrictions: ____________________________
Restrictions: ____________________________
𝑥−7
23. Sketch the graph of 𝑦 = 𝑥 2−3𝑥−28
Holes:
Vertical Asymptote:
Horizontal Asymptote:
Domain:
Range:
Data and Statistics
24. You collect data from your friends on the number of
days they have been absent this year from school:
0
0
2
3
3
3
4
6
8
12
25. The mean weight the classes backpacks is 18.4 lbs; the
standard deviation is 3.1 lbs. All the data values are
within three standard deviations of the mean.
a) What is the maximum value of the data?
b) Within how many standard deviations of the mean
is 13 lbs.?
a. Find the mean and the standard deviation of the
data. Round to the nearest tenth place.
b. How many values in the data set fall within one
standard deviation of the mean?
c. Within two standard deviations?
26. The transportation department collected data on bike
27. The average height of the a women is 64 inches (5 ft 4
riders and found the average speed for riding a bike
was 9.6 mph. If the data showed a normal distribution
with a standard deviation of 1.8, how many bicycler
riders would you expect out of 200 to be riding at a
speed less than 6 mph?
in.) with a standard deviation of 3 inches. If the data is
normally distributed, what percent of women are over
67 inches tall (5 ft 7 in.)?
28.
Determine whether each of the questions is biased or unbiased and explain why.
a) Would you prefer early release or late start on
shortened school days?
Biased or Unbiased? Why?
b) Do think teachers have the right to assign detentions
for late homework?
Biased or Unbiased? Why?
29. Your math teacher samples all her of 150 students, 30 forgot to do their homework last night.
a. Find the sample proportion for those who forgot to do their homework.
b. Calculate the margin of error.
c. Use the margin of error to calculate the true population proportion who forgot to do their homework.
Trigonometry
Find the following values.
7𝜋
30. Sin ( 4 ) =
31. tan(−270°) =
32. Find the amplitude, vertical shift, phase shift, period of
the function. Then sketch one cycle of the graph of
each function.
𝜋
𝑦 = 2 cos (𝑥 − ) − 1
2
Amplitude = __________
Midline = __________
Phase Shift = __________
Period = __________
Frequency=_________________
33. Write a sine and cosine equation for the following
34. Graph: 𝑦 = tan(𝑥) + 2
graph.
Sine Equation: ______________________________
Cosine Equation: ____________________________
35. Graph 𝑦 = 3 sin(𝑥) − 1
36. a) Sketch −120°
b) Find a positive and negative coterminal angle with
−120°
Answer Key:
1.
5.
Domain:
𝑥 ≥ −5
Range:
𝑦 ≥ −2
left 5, down 2
2.
Domain: all real #s
Range: all real #s
𝑔−1 (𝑥) = (𝑥 + 2)2 − 5
right 6, down 1
ℎ−1 (𝑥) = (𝑥 + 1)3 + 6
√6
2
6.
√42𝑥
7
3
4
3. 4𝑠𝑧 3 √𝑠 3
4. 5𝑢𝑣 2 √5𝑢2 𝑣 2
7. 3𝑥𝑦 2 √𝑥
8. 𝑥 = 1.5
9. 𝑛𝑜 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛
10. Use calc to check
11. Use calc to check
12. 𝑥 = 80
13. 𝑥 = 2
17. 22.4 years
14. 4
18. 7445.8 years
15. 0.67
21. 𝑥 = 20.33
No restrictions.
22. 𝑥 = −7, 𝑥 = 1
Restrictions: 𝑥 ≠ −4
25. A) 27.7 lbs
B) 2 standard deviations
29. A) 20%
B) ±8.1%
C) 11.9% − 28.1%
26. 5 riders
23. Hole: 𝑥 = 7
VA: 𝑥 = −4
HA: 𝑦 = 0
Domain: 𝑅, 𝑥 ≠ −4, 7
Range: 𝑅, 𝑦 ≠ 0
27. 16%
16. −2
20. Hole: 𝑥 = 2
VA: 𝑥 = 7
HA: 𝑦 = 4
Domain: 𝑅, 𝑥 ≠ 2, 7
Range: 𝑅, 𝑦 ≠ 4
24. A) mean: 4.1, SD is 3.5)
B) 8
C) 9
33. 𝑦 = 2 sin(𝑥) − 3
34.
𝜋
𝑦 = 2 cos (𝑥 − ) − 3
2
30. −
√2
2
19. 𝑦 =
3
−
(𝑥+4)
31. undefined
35.
3
28. A) unbiased
B) Biased
32. Amplitude = 2
Midline= -1
𝜋
Phase Shift = 2
Period = 2𝜋
Frequency= 1
36. 240° and −480°