Precalculus: 2nd Semester Review
NAME _________________________________________
** Marks calculator OK
Sections 4.4, 4.5, 4.7 (Trig Graphs, Inverse Trig)
1. Sketch the following graphs and describe their characteristics:
a. 𝑦 = 2 − 4𝑐𝑜𝑠𝑥
b. 𝑦 = 5𝑡𝑎𝑛2𝑥
𝜋
𝜋
2. For the equation 𝑦 = −5 sin 3 (𝑥 + 4 ) − 9, find the amplitude, vertical shift, period and phase shift.
3. A Ferris wheel 32 feet in diameter makes one revolution every 40 seconds. If the bottom of the wheel is 6
feet above the ground, write the equation to model the path of the rider on the wheel. Then, use the equation
to find the first three times a rider is 25 feet above the ground (assume he gets on at the bottom).**
4. Find the following (give answers in radians):
a. sin−1 (
−√3
2
)
b. cos−1 (
−√2
2
)
c. tan−1(−√3)
Chapter 5 (Trig Identities, Solving Trig Equations)
5. Simplify the following:
a.
𝑐𝑜𝑠2 𝑥+𝑠𝑖𝑛2 𝑥
𝑡𝑎𝑛𝑥
b.
1+𝑡𝑎𝑛2 𝑥
𝑐𝑠𝑐 2 𝑥
6. Solve the following over [0,2𝜋):
a. 1 − √2 cos 𝑥 = 0
b. 2𝑠𝑖𝑛2 𝑥 + 𝑠𝑖𝑛𝑥 − 1 = 0
d. cot −1 (−1)
c. 𝑠𝑖𝑛2𝑥 − 𝑐𝑜𝑠𝑥 = 0
7. Prove the following:
a. csc x  sin x  cot x cos x
c.
sin x
 csc x  cot x
1  cos x
d. 3𝑡𝑎𝑛2 𝑥 − 1 = 0
b. cot x(tan x sin x  cos x)  csc x
d. cos 3 x  cos3 x  3sin 2 x  cos x
Chapter 6 (Vectors, Parametric Equations, Polars, Complex Plane)
8. Convert the following coordinates from polar to rectangular or from rectangular to polar:
c. (-8,6)**
d. (5,-4)**
b. (−8,265°)**
a. (3,30°)
9. At the driving range, a golf ball is struck from an elevated tee box 8 feet above the ground. The ball was
struck with an initial velocity of 180 ft./sec. at an angle of 38° to the horizontal.**
a. Write the parametric equations for the flight of the ball.
b. Find when and where the ball will be at its
highest point.
c. Determine when and where the ball will
strike the ground.
10. Find the component form of a vector with a magnitude of 350 and a direction angle of 140°.∗∗
11. Find the direction angle and magnitude of a vector with a component form ⟨−90, 68.5⟩.**
12. Graph the following polar equations.
a. r  2  3sin 
a. r  2  2cos
13. Write the equation to match the polar described below:
a. Write the equation of a limaςon that has xintercepts at 4 and -4, and y-intercepts at 5
and -3.
14. Eliminate the Parameter: x = 2t2 + 3; y = t – 1
b. r  5sin  2 
b. r  3cos
b. Write the equation of a circle that lies on the
negative y-axis, with y-intercepts of 0 and
-10.


4
15. Use De Moivre’s Theorem to simplify 2 3  2i . Express your answer in standard form.
Chapter 7 (Matrices)
16. Use the matrices A, B, C, and D to find the following:
3 −2
𝐴=[
]
4 2
4 −2
𝐵=[
]
−6 3
−1 0
𝐶=[ 3
5]
4 −2
6
𝐷 = [4
5
a. 2𝐴 − 3𝐵
b. 𝐶 × 𝐵
c. 𝐴 × 𝐶
d. 𝐷 × 𝐶
e. 𝐵 −1
f. The determinant of 𝐴
3 −8
−2 −1]
3
1
17. Solve the system using the indicated method**: 2𝑥 − 3𝑦 + 𝑧 = 14
3𝑥 − 4𝑦 + 6𝑧 = 29
𝑥 + 2𝑦 − 5𝑧 = −11
a. Inverse Matrices
b. Reduced Row Echelon Form
Conics
18. Convert the following parametric equation of a conic section into general form:
𝑥 = 2 + 3 cos 𝑡
𝑦 = 4 + 6 sin 𝑡
19. Determine the shape of the conic section and identify all important information before graphing. (If
necessary, convert the following equations into general form by completing the square.)
a. 𝑥 2 − 6𝑥 + 𝑦 2 + 8𝑦 − 11 = 0
b. 𝑦 2 + 4𝑦 − 4𝑥 − 8 = 0
c.
(𝑦−3)2
9
-
(𝑥+2)2
25
=1
d. 𝑥 2 + 2𝑥 + 2𝑦 2 − 8𝑦 = 7
20. Find the equation for a parabola with focus at (7, 2) and directrix x = -1.
21. Find the equation of an ellipse with foci at (-3, -2) and (1, -2) and major axis length = 8.
Chapter 9 (Combinatorics, Permutations and Probability)
22. At a Wendy’s restaurant a customer can order a single, double or triple hamburger with up to 12 condiments
(ketchup, mustard, onions, pickles, etc.). How many different ways can the customer order a burger?**
23. Write the sample space for all outcomes of flipping 3 different coins.
24. How many distinguishable 9 letter “words” can be formed using the letters in TENNESSEE?**
25. How many ways can 9 starters be chosen from a team of 20 baseball players if each has equal abilities?**
26. 17 baseball players show up to a game. How many ways can the manager write out a batting order of 9
batters?**
27. Assume that the probability of giving birth to a boy or a girl is the same.
a. In a family of 3 children, what is the probability of:
i. Having only one girl?
ii. Having either all boys or all girls?
b.
In a family of 4 children, what is the probability of:
i. Having three girls and
ii. Having at least three
one boy?
girls?
iii.
Having 2 of each?
28. The probability of a colored gum ball coming out of a machine is shown in the table below.**
RED WHITE BLUE GREEN YELLOW
0.25
0.33
0.18
0.16
0.08
If one gumball is purchased, find each probability:
a. P(White)
b. P(Blue or Green)
If two gumballs are purchased, find each probability:
a. P(both Red)
b. P(Green then Yel.)
c. P(Blue and Green)
c. P(Not Yellow)
d. P(No White)
29. Johnny’s batting average against Pitcher X is .365. Pitcher X has a 40% chance of pitching today. Find the
probability that:**
a. Johnny gets a hit off Pitcher X today.
b. Pitcher X makes an out from Johnny’s atbat.