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GPS Pre-Calculus Final Exam Study Guide Unit 3 Rational Functions 1. Give the degree, the number of real zeros, the number of (nonreal) complex zeros, for this polynomial function.

• Degree: _________________________________

• Real zeros: ______________________________

• Complex zeros: __________________________

2. Sketch the graph of

.is

3. The function

( ) .is what transformation of

in Problem 2?

4. What kind of discontinuity does the graph in Problem 3 have at x = –2?

5.Simplify and then sketch the graph of ( )( )

( )

What kind of discontinuity will the graph have at x = 3?

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6. This cubic function has three zeros, although only two of them seem to show up on the graph. Write the three zeros.

Rational Function Problem: The figure shows the graph of the rational function

5 13

( 1)( 5)

xy

x x

7. Explain why the discontinuities at x = 1 and x = 5 are vertical asymptotes rather than removable discontinuities.

Unit 5: Graphs of Trigonometric Functions

Graph each function.

1. y = 3 sin x + 2

2. y = -2cos (2x - )

3. y = tan

22

1 x

Write the equation of the following graphs.

1.

X

Y

-6.28 -3.14 3.14 6.28

-5-4-3-2-1

12345

0

X

Y

-6.28 -3.14 3.14 6.28

-5-4-3-2-1

12345

0

3

Write the equation of a cosine function with the following conditions.

6. amplitude: 4, period: 2

, phase shift:

4

7. amplitude: 2, period: 2, phase shift: -1, vertical shift: -3

Graph each function.

8. y = 2cot

x

2

3+ 2

9. y = 3sec

x

3

1

10. y = -3sin

3

x

Find the following.

___________ 11. sin-1

2

3 __________ 12. arccos (0)

__________ 13. tan-1 (1) __________ 14. sin-1

2

2

__________ 15. sin

2

3arccos __________ 16. cos 3tan 1

+ arcsin (-1))

Complete the chart with exact values.

17.

Period Amplitude Phase Shift Vertical Shift

y = -8cos(4x - ) + 2

y = 5sin(3x + 2) - 3

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18.

Period Amplitude Phase Shift Vertical Shift

y = 3tan(3x) + 1

y = 2cot(2x) + 2

19.

Period Amplitude Phase Shift Vertical Shift

y = 2sec(4x) + 1

y = -3csc

x

2

1+ 3

Unit 1 Data Analysis

I. Multiple Choice. Choose the one best answer.

1. In a large population of adults, the mean IQ is 130 with a standard deviation of

15. Suppose 100 adults are randomly selected for a market research

campaign. The distribution of the sample mean IQ is

A) exactly normal, mean 130, standard deviation 20.

B) approximately normal, mean 130, standard deviation 0.1.

C) approximately normal, mean 130, standard deviation 1.5.

D) approximately normal, mean 130, standard deviation 20.

E) exactly normal, mean 130, standard deviation 1.5.

2. The central limit theorem is important because it allows us to use the Normal distribution to make inferences about the population mean

A) if the sample size is reasonably large B) if the population is Normally distributed and the sample size is reasonably large C) if the population is Normally distributed D) if the population is Normally distributed and the population variance is known E) if the population size is reasonably large

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3. A 95% confidence interval for the mean math scores for a population of fifth-graders is

(43.6, 58.9). If you computed a 90% confidence interval using the same information, which of the following statements is correct?

A) The intervals have the same width. B) The 90% confidence interval is wider. C) The 90% confidence interval is shorter. D) The margin of error can’t be found. E) The answer can’t be determined from the information given.

4. The weights of 12 men have mean 180 pounds and standard deviation 15 pounds. What is the standard error of the mean?

A) 3.098 B) 4.330 C) 4.648 D) 5.196 E) Cannot be determined with information given

5. If all other parts of problem stay the same, which of the following would reduce the width of a confidence interval for a given population.

I. Increase the sample size. II. Have a smaller sample standard deviation. III. Increase the confidence level.

A) I only B) II only C) III only D) I and II only E) I, II, and III

II. Free Response questions. Answer completely, but be precise. 6. Consider the sampling distribution of sample means obtained by random sampling from an

infinite population. This population has a distribution that is highly skewed toward the larger

values.

a. How is the mean of the sampling distribution related to the mean of the population?

b. How is the standard deviation of the sampling distribution related to the standard deviation of the population?

c. How is the shape of the sampling distribution affected by the sample size?

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7. A random sample of 120 households in a certain affluent community yields a

mean weekly food budget of $120 and a standard deviation of $15. Find the 95%

confidence interval for weekly food budgets of households in this community.

8. It is believed that the average amount of money spent per US household per

week on food is about $115. Is the community in problem number 9 representative

of the US household? Justify your answer.

Unit 4 : Trigonometric Functions

Change between Radians and Degrees

1. 55o 2.

3. 315o 4.

Tell the quadrant the terminal side of the given angle would be.

5.

6. 325o 7.

8. -522o

Find one positive and one negative co-terminal angle with the given angle.

9. 650 10.

11. If 5

sin9

what would Cos θ be?

12-14. Find the missing sides and angles of the given triangle.

A 14 C

A = ____ a = _____

B = 56o b = 14

C = 90o c = _____

56o

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Draw the following angles in standard position, find the reference angles and mark them.

15. 122.52’ 16.

Use the given point on the terminal side of an angle x in standard position. Evaluate the six

trigonometric functions of x.

17. (-6, 8)

Sin x = Csc x = cos x = Sec x = Tan x = Cot x =

Given a trig ratio of an angle x and the quadrant the terminal side of the angle is in, evaluate the

remaining 5 trigonometric functions of x.

18. Sin x =

and cos 0 .

Sin x =

Csc x =

Cos x = Sec x =

Tan x = Cot x =

19. If cot θ = 5.67128, find θ in degrees.

A) 79.889 B) 27.38 C) 10 D) .0337 E) .0090

Unit 2 Sequences and Series

1. Show that you know the difference between a sequence and a series by writing the first three terms of a geometric sequence and a geometric series, each of which has a first term 5 and common ratio 4.

Sequence:

Series:

2. Show that the sequence 2, 8, 24, … is neither arithmetic nor geometric.

3. For the following sequence: 4, -16, 64, … a. Find a possible formula for tn b. Find t21

4. Determine the sum of the series: 93 + 89 + 85 + … + 5

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5. What is the 11th term of the sequence an = 3n – 5?

6. What is the sum of the first five terms of a if an = 2n2?

7. What is the sum of the first six terms of the sequence an = 2n + 3?

8. What is the sum of the first 45 terms in the arithmetic sequence a1 = 3; an = an-1 + 6

9. Calculate:

∑

10. Calculate:

∑ ( )

11. What is the limit of the infinite series: 3 + 1 + 1/3 + 1/9 …. 12. An arithmetic series has t1 = 12 and t2 = 23.

a. Find the common difference

b. Find t200.

c. Find S200.

d. Find n if tn = 3367.

e. Find n if Sn =23985.

13. A geometric series has t1 = 50 and t2 = 40. a. Find the common ratio.

b. Find t30.

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c. Find n if tn = 10.48576… (Hint: solve the exponential equation using logarithms)

d. Find S3.

e. Find the limit of the infinite series Sn.

14. Assume that this year, n = 1, the average annual salary for women in the United States is $31,000 and it is increasing by 5% per year, meaning that each year the salary is 1.05 times what it was the previous year. Assume that the average annual salary for men is $35,000 and that it is increasing by $1,500 per year.

a. Write the first three terms of each sequence of salaries. What kind of sequence is each?

b. In the first three years, whose salaries are increasing faster, women’s or men’s?

c. Assuming that the same pattern of salaries continues, who will be making more, women or men, in year n = 25?

15. The student council is going to sell tickets for a candy-give-away for $5. All the tickets will be drawn, with the first person receiving 1 piece of candy, the second person receiving 2 pieces, the third 4 pieces, the fourth 8 pieces and so on. The student council expects to sell 20 tickets.

a. How many pieces of candy will the 20th person receive?

b. How many pieces of candy will they need for all 20 tickets?

c. If the cost for the candy is $1 per 1,000 pieces, what will be the profit/loss for the fundraiser?

Unit 6: Identities

Find the following using trigonometric identities. Assume is in

quadrant I.

1. If sec = 4, find tan . 2. If cot =3

1, find sin .

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Simplify.

3. cos2x tan x csc x 4. x

xx2tan1

csctan

Prove the following identities

5. cos2β∙tan2β + cos2β = 1 6. xx

x 2

2

2

tancot1

tan1

7.

csc

tan

sec 8. AA

A

Atansin

cot

cos2

Solve for values of such that 0 < 360.

9. 3cos2 - 1 = 0 10. 2sec θ + 2 = 0

11. 4sin2θ = 4sin - 1 12. 2cos2 = cos

Find the exact value using sum/difference or half-angle identities.

13. sin 15 14. cos 195

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If α and β are measures of two first quadrant angles, find the exact value of each function.

15. If sin α =3/5 and cos β = 5/13, find sin (α + β).

16. If cos α =24/25 and sin β =8/17, find cos (α – β).

Unit 7: Law of Sines / Law of Cosines / Heron’s Formula

Solve the triangle.

1. C = 72o a = 4 b = 5 2. C = 135o b = 8 c= 14

3. A = 115o a = 14 b = 16

4. C = 115o b = 95 c = 105

Use Heron’s formula to find the area of the triangle.

5. a = 6 b = 8 c = 12

6. C = 120o a = 10 b = 14

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Solve each problem. Draw any diagrams necessary and show all work.

7. Because of the prevailing winds, a tree grew so that it was leaning 8o from the vertical. At a point 100 feet away from the tree, the angle of elevation to the top of the tree is 34.1o. Find the height of the tree.

8. The angles of elevation to an airplane from two points A and B on level ground are 53o and 63o respectively. The points A and B are 7 miles apart, and the airplane is between those positions. Find the altitude of the plane. (Hint: Find one side of the triangle you draw and then use that side to be the hypotenuse of a right triangle with height as the altitude)

9. Find the area of triangular parcel of land if its sides are 500 ft, 600 ft and 900 ft.