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SKILL REVIEW
Below are skills that you will need in order to do well on Chapter 2. You are responsible for the information on
this sheet. If you have any questions you need to come see me in ac lab or before/after school.
Concept #1: Factoring!
Factor the following:
1. x 2  9 x  20
2. x 2  3 x  12
3. x 2  100
4. y 2  16 y  64
5. 3 x 2  20 x  7
6. 2w2  w  3
7. 4 x 2  9 x  2
8. 9 x 2  64
9. 8 x3  27
10. 64 x 3  1
11. x3  7 x 2  10 x
12. x 3  7 x 2  9 x  63
Concept #2: Quadratic Formula.
Use the quadratic formula to solve for x.
13.
x2  4x  5  0
14. 8w2  8w  2  0
15. 6  2t 2  9t  15
17. 450.4  7
18. 678235  42
Concept #3: Long Division
16. 5467  32
Concept #4: Evaluating Functions
Evaluate the following functions
2
19. f ( x)  x  5x  3
a) f(1)
b) f(-2)
c) f(0)
3
2
20. g ( x)  x  x  x
a) f(-1)
b) f(3)
c) f(y)
Chapter 2: Polynomial Functions
Essential Questions
What is a polynomial?
Why do we study polynomials as a group?
Why is it important to be able to find the roots/zeros/x-intercepts of a polynomial?
What is an imaginary number?
Why do we study imaginary numbers?
What does it mean for a function to have imaginary roots/zeros?
Why is it important to be able to model a set of data using a function?
Learning Targets
Determine whether a function is a polynomial.
Find the complex roots/zeros/x-intercepts of a polynomial.
Graph all polynomials.
Write all polynomials as a product of their linear factors.
Determine what degree polynomial should be used to model different sets of data.
Perform operations on complex numbers.
Homework
Section 2.1 – Quadratic Functions
Describe, in words, the transformations that are occurring to the graph of y  x 2 .
1. (a ) y 
(c ) y 
3 2
x
2
(b) y 
3
2
 x  3
2
3 2
x 1
2
(d ) y  
3
2
 x  3  1
2
2. (a) y  4 x 2
(b) y  4 x 2  3
(c) y  4( x  2)2
(d ) y  4  x  2   3
2
Sketch the graph of the quadratic function. Identify the vertex and x-intercept(s).
4. f ( x)  16 
3. f ( x)  x 2  7
5. f ( x)   x  6   3
2
1 2
x
4
6. f ( x)  x 2  10 x  14
7. f ( x)  4 x 2  24 x  41
Write the equation in standard form of the quadratic function that has the indicated vertex and travels through
the given point.
8. Vertex: (4, -1) and Point: (2, 3)
 5 
 7 16 
9. Vertex:   , 0  and Point:   ,  
3
 2 
 2
10. A rancher has 200 feet of fencing. He plans on enclosing a rectangular area and then dividing that area into
two congruent pens by putting a fence in the middle. What are the dimensions of the enclosed rectangular area
that will produce the maximum area?
4
24
x  12 where y is the height in feet and x is the horizontal
11. The path of a diver is given by y   x 2 
9
9
distance in feet from the end of the diving board. What is the maximum height of the diver?
Section 2.2 – Polynomial Functions of Higher Degree
For exercises 1-8, match the polynomial functions with its graph.
1. f ( x)  2 x  3
(a)
(b)
(c)
(d)
(f)
(g)
2. f ( x)  x 2  4 x
3. f ( x)  2 x 2  5x
4. f ( x)  2 x3  3x  1
1
5. f ( x)   x 4  3x 2
4
1
4
6. f ( x)   x3  x 2 
3
3
7. f ( x)  x 4  2 x3
8. f ( x) 
1 5
9
x  2 x3  x
5
5
(e)
Graph the functions. Make sure to label their roots.

1 3 2
x x 9
4

9. f ( x)  5x 2  10 x  5
10. f ( x) 
11. f ( x)  x5  6 x3  9 x
12. f ( x)  5x 4  15x 2  10
13. f ( x)  x 4  4 x 2
14. f ( x)  3x3  24 x 2
Find a polynomial function that has the given zeros.
15.  7, 2
16. 0, 2, 5
17.  2,  1, 0, 1, 2
18. 6  3, 6  3
19. An open box is to be made from a square piece of material 36 centimeters on a side by cutting equal
squares with sides of length x from the corners and turning up the sides. Find the formula for the volume of the
box and determine the dimensions of the box that will maximize the volume.
Section 2.3 – Real Zeros of Polynomial Functions
Use synthetic division to divide the polynomials.
1. (5x3  18x 2  7 x  6)  ( x  3)
2. (2 x3  14 x 2  20 x  7)  ( x  6)
3. (5x3  6 x  8)  ( x  2)
4. ( x4  3x 2  1)  ( x 2  2 x  3)
5. ( x3  729)  ( x  9)
Use synthetic division to find each function value.
6. f ( x)  4 x3  13x  10
(a) f (1)
(b) f ( 2)
1
(c ) f  
2
(d ) f (8)
(b) f ( 4)
(c) f  3
(d ) f (1)
7. f ( x)  x6  4 x 4  3x 2  2
(a ) f (2)
Find all of the roots of the given polynomials. Write each function as the product of linear factors.
8. f ( x)  x3  4 x 2  4 x  16
9. f ( x)  4 x5  8x 4  5x3  10 x 2  x  2
10. f ( x)  x3  12 x 2  40 x  24
11. f ( x)  6 x 4  11x3  51x 2  99 x  27
Find all of the solutions of the polynomial equations.
12. x 4  x3  29 x 2  x  30  0
13. x5  x4  3x3  5x2  2 x  0
Section 2.4 – Complex Numbers
Perform the given operation and write your answer in standard form.

 
1. (11  2i)  (3  6i)
2. 7  18  3  3i 2
3. 22  (5  8i)  10i
3 7  5 1 
4.    i     i 
4 5  6 6 

5.   3.7  12.8i   6.1  24.5
7.

75

2
9.  3i(6  i)

6. 5  10
8. (6  2i )(2  3i)


10. 3  5 7  10


12. 
11. (1  2i)2  (1  2i)2
5
2i
14.
8  7i
1  2i
2
16.
2i
5

2i 2i
17. 4i 2  2i 3
18.

13.
15.
3
1 i
5i
 2  3i 
2

6
Section 2.5 – The Fundamental Theorem of Algebra
Find all zeros of the functions.
1. f ( x)  x 2 ( x  3)
2. f ( x)  ( x  4)3 ( x  2)
3. f ( x)  ( x  9)( x  2i )( x  2i )
4. f ( x)  ( x  3)( x  2)( x  3i)( x  3i)
Find all of the zeros of the functions. Then graph them and write the functions as the product of linear factors.
5. f ( x)  x3  11x 2  39 x  29
6. f ( x)  3x3  4 x 2  8 x  8
7. f ( x)  x 4  29 x 2  100
8. f ( x)  x 4  6 x3  10 x 2  6 x  9
9. f ( x)  x3  10 x 2  33x  34
Find a polynomial with real coefficients that has the given zeros.
10. 4, 3i,  3i
11.  1, 6  5i
12. 0, 0, 4, 1  i 2
Use the given root to find all of the roots of the function.
13. f ( x)  x3  x 2  9 x  9, Root  3i
14. f ( x)  4 x3  23x 2  34 x  10, Root  3  i
15. f ( x)  x3  4 x 2  14 x  20, Root  1  3i
Chapter 2 Review
For 1-3, sketch the graph of the quadratic functions. Make sure to label the vertex and any intercepts.
1. f ( x)  3x 2  12 x  11
3. f ( x)  
2. f ( x)  x 2  2 x  3
1
2
 x  1  3
2
4. f ( x)  x 4  2 x3  15x  2 Find f (2).
5. f ( x)  2 x5  5x 4  8x  20. Find f
 2 .
Write the complex number in standard form.
7. 5i 2 (13  8i)
6.  12  3
8.
1  7i
2  3i
9. 10  8i  2  3i 
Find all of the zeros of the given polynomials.
10. f ( x)  x3  4 x 2  x  6
11. f ( x)  x 4  11x3  41x 2  61x  30
Sketch the graph of the polynomial functions. Make sure to label any intercepts and their multiplicities.
12. f ( x)  x  x  2  x  5 
13. f ( x)   x  1  x  1  x  3
2
3
14. f ( x)  x 4  10 x3  26 x 2  10 x  25
15. f ( x)  2 x3  x 2  x
16. Find the polynomial function with the given zeros.
a) 1, -4, -3i
b) -2i, 3
c) 1+3i, 0, 0, 2
d) -3-i, 4, 1, 1
Write an equation that represents the graph below.
17. f ( x) 
Match the polynomial function to the given zeros and multiplicities.
a)
b)
c)
d)
4
2
18. -3 (multiplicity of 2), 2 (multiplicity of 3)
19. -3 (multiplicity of 3), 2 (multiplicity of 2)
20. -1 (multiplicity of 4), 3 (multiplicity of 3)
21. -1 (multiplicity of 3), 3 (multiplicity of 4)
22. A plumber’s total charge includes a fixed service charge plus an hourly rate for the job. If the total charge is
$140 for a 3-hour job and $200 for a 5-hour job, what is the total charge for an 8-hour job?
a. $430
b. $290 c. $260
d. $250
e. $240
23. A function that is defined by the set of ordered pairs {(2,1), (4,2), (6,3)} has domain {2, 4, 6}. What is the
domain of the function defined by the set of ordered pairs {(0,2), (2,2), (3, -2)}?
a. {2}
b. {-2, 2}
c. {-2, 0, 3}
d. {0, 2, 3} e. {-2, 0, 2, 3}
24. Which of the following expressions is equivalent to 3x4 + 6x2 – 45?
a. (x2+5)(x2-3)
d. 3(x2-5)(x2+3)
b. (3x2-15)(x2+3)
e. 3(x2+5)(x2-3)
c. 3(x4+6x-45)
25. What is the sum of the 2 solutions of the equation x2+x-6=0?
a. -6
b. -3
c. -1
d. 0
e. 2
SKILL REVIEW
Concept #1:Solving Systems
Solve the following system using the given method:
4x – 3y = 8
8x – 6y = 16
1. Solve using graphing
2. Solve using substitution
3. Solve using linear combinations
Chapter 7: Solving Systems of Equations
Essential Questions
What is a system of equations?
What does a solution to a system of equation represent in terms of its equations? Its graphs?
Which method do you prefer when trying to solve a system of equations? Why would you choose one method
over the other?
Learning Targets
Solve systems of equations using substitution, linear combination, and graphing.
Homework
Section 7.1 – Solving Systems of Equations
Solve the systems of equations using substitution.
1.
x  2y 1
5 x  4 y  23
2.
6x  3y  4  0
x  2y  4  0
.5 x  3.2 y  9
3.
.2 x  1.6 y  3.6
1
3
x  y  10
4
4. 2
3
x y 4
4
2x  y  4
5.
4 x  2 y  12
2
 x y 2
6. 3
2x  3 y  6
7.
y  x
y  x  3x 2  2 x
9.
x  2 y  1
x y 2
3
8.
x y 0
3 x  2 y  10
10. A small fast-food restaurant invests $5000 to produce a new food item that will sell for $3.49. Each item
can be produced for $2.16. How many items must be sold to break even?
Section 7.2 – Systems of Linear Equations in Two Variables
Solve the systems of equations using elimination.
2x  5 y  8
1.
5 x  8 y  10
2
1
2
x y 
2. 3
6
3
4x  y  4
1
1
x  y 1
3. 4
6
3x  2 y  0
x 1 y  2

4
4. 2
3
x  2y  5
5.
.05 x  .03 y  .21
.07 x  .02 y  .16
6.
7 x  3 y  16
y  x 1
7.
y  3 x  8
y  15  2 x
8.
4x  3y  6
5 x  7 y  1
9. Find a system of equations that has (3, -4) as a solution.
10. 600 tickets were sold at the last 5FD show. The tickets for adults cost $5 and the tickets for students cost
$3. If the receipts for the show totaled $2330, how many of each type of ticket was sold?
Section 7.3 – Multivariable Linear Systems
Solve the systems of equations.
x yz 2
1.
4x  y  4
 x  3 y  2z  8
2 x  4 y  z  4
3. 2 x  4 y  6 z  13
4x  2 y  z  6
4 x  y  3z  11
2. 2 x  3 y  2 z  9
x  y  z  3
5x  3 y  2 z  3
4. 2 x  4 y  z  7
x  11y  4 z  3
2 x  y  3z  1
5. 2 x  6 y  8 z  3
6 x  8 y  18 z  5
2 x  y  3z  4
6.
4 x  2 z  10
2 x  3 y  13z  8
x  4 z  13
7.
4x  2 y  z  7
2 x  2 y  7 z  19
x  3y  z  4
8. 4 x  2 y  5 z  7
2 x  4 y  3z  12
3x  2 y  6 z  4
9. 3x  2 y  6 z  1
x  y  5 z  3
11.
10.
12 x  5 y  z  0
23x  4 y  z  0
10 x  3 y  2 z  0
19 x  5 y  z  0
An object moving vertically is at the given heights at the given times. Find the position equation
s
1 2
at  v0t  s0 for the object.
2
12. At t  1 sec, s  128 ft
At t  2 sec, s  64 ft
13. At t  1 sec, s  132 ft
At t  2 sec, s  100 ft
At t  3 sec, s  48 ft
At t  3 sec, s  36 ft
Chapter 7 Review
Solve the systems of equations.
1.
3.
x  3 y 2  13
2 x  y 2  2
x2  y 2  4
x y 0
2.
4.
3x  4 y  0
2 x  3 y  17
x2  y 2  2
x2  y 2  0
5.
 x  5 y  13
2 x  9 y  24
6.
x  7 y  8
11x  13 y  2
2 x   y 2  3
7.
4 x  2 y 2  6
3x  2 y  4 z  5
8. 2 x  3 y  z  9
 x  8 y  6 z  13
2 x  y  4
9. x  4 y  2 z  3
3x  4 y  z  2
4 x  y  2 z  10
10. 2 x  y  3z  6
2 x  z  7
11. Mr. Wu spent his three day weekend selling lemonade on the streets. He spent $30 on signs, a chair, and a
table for his stand. If he sold each cup of lemonade for $.25, but it cost $.09 total for the cup, ice, lemons, and
sugar, how many cups of lemonade would Mr. Wu have to sell to break even?
12. Mr. Coulson has $5000 to invest. He splits the money into an IRA and a 403b, which have a return of 3% and
6% respectively. If he earns $240 total after one year, how much did Mr. Coulson invest into each account?
13. Mrs. Hopkins, Mr. Coulson, and Mr. Wu went to six flags together this summer. They bought three tickets,
six cheeseburgers, and two jumbo ice cream cones for a total of $140. One ticket cost as much as all of the
cheeseburgers, and you could buy 2 cheeseburgers for the cost of one jumbo ice cream cone. How much does a
jumbo ice cream cone cost?
14. A plumber’s total charge includes a fixed service charge plus an hourly rate for the job. If the total charge is
$140 for a 3-hour job and $200 for a 5-hour job, what is the total charge for an 8-hour job?
a. $430
b. $290
c. $260
d. $250
e.$240
15. As a fund-raiser, a local youth group sold boxes of regular popcorn for $5 each and boxes of caramel popcorn
for $8 each. Altogether, they sold 160 boxes for $1,100. How many boxes of caramel popcorn did they sell?
a. 20
b. 32
c. 60
d. 80
e. 100
16. On a recent test, some questions were worth 2 points each and the rest were worth 3 points each. Tuan
answered correctly the same number of 2-point questions as 3-point questions and earned a score of 60 points.
How many 3-point questions did he answer correctly?
a. 36
b. 30
c. 20
d. 12
e. 10
17. Becky has 76 solid-colored disks that are either red, blue, or green. She lines them up on the floor and finds
that there are 4 more red disks than green and 6 more green disks than blue. How many red disks does she
have?
a. 10
b. 20
c. 24
d. 26
e. 30
Midterm Review
1) Describe and graph the transformation of the function f(x)= -1/3 (x-4)2 + 5.
2) Find the polynomial of degree 4 with zeros at 4, 0, and -2i. Multiply and simplify your answer.
3) Determine the intervals on which the function f ( x)  x 4  12 x3  6 x is increasing and decreasing, and find
the relative maximum and minimum.
4) Find an equation of the line that passes through the point (0, 4) and is (a) parallel to and (b) perpendicular to
the line 5 x  2 y  3.
5) Find the domain of: f ( x)  10  3  x .
For problems 6-8 (a) identify the parent function of f, (b) describe in words the transformations from f to g, and
(c) sketch a graph of g.
6) g ( x)  2( x  5)3  3
8) g ( x)  4  x  7
7) g ( x)   x  7
9) Determine whether the function has an inverse, and if so, find the inverse function.
b) f ( x) 
a) f ( x)  x3  8
3x 2
8
10) Divide x4 + 3x2 – 9x + 5 by (x + 3).
11) List the possible rational zeros of the function. Use a graphing calculator to graph the function and find all
the real zeros.
a) g ( x)  2 x 4  3x3  16 x  24
b) h( x)  3x5  2 x 4  3x  2
12) Perform the operation and write the result in standard form.
a) (8  3i )  (1  15i)
b) (4  3i)  (5  i)
c) (2  i )(6  i )
d)
3i
7i
13) Write an equation for a polynomial function with zeros at 6, 2(multiplicity of 2), 0, and -3. Draw a graph of
the polynomial. State the degree of the polynomial.
14) Find the zeros of the graph and state their multiplicity, then write an equation for the graph. Multiply and
simplify your answer.
Solve the following problems without a calculator. Check your answer by solving with a calculator.
15)
x 2  y 2  61
x  y 1
x yz 9
18)  2 x  y  z  3
x  y  1
16)
x3  y  3
y  x2  3
7 x  5 y  24
19) 1
1
1
 x y 
3
2
3
4x  2 y  z  7
21) 6 x  3 y  2 z  14
2 x  5 y  3z  8
2 x  3 y  4 z  16
22) 3 x  2 y  z  6
x y 5
x  4 z  13
24) y  2 z  6
3 x  2 y  4 z  15
2 x  5 y  19 z  34
25) 3 x  8 y  31z  54
x  3 y  12 z  20
17)
3 x  2 y  5
5 x  y  6
3x  y  2 z  6
20) 5 x  y  3z  7
2 x  3 y  4 z  5
23)
1.3x  1.7 y  4
2 x  y  30
26) Mr. Coulson likes to mix up his own fertilizer to keep his yard looking green and pristine. As we all know,
fertilizer is made up of Nitrogen (N), Phosphorus (P) and Potassium (K). The proportion of each component
for a given type of fertilizer is shown on the label as three numbers in N-P-K order. For example, a 5-10-15
fertilizer has 5/30 or 1/6 N, 10/30 or 1/3 P and 15/30 or 1/2 K. Farmer C. wants to put down 120 lbs of 125-7 fertilizer, but only has bags containing 10-10-10 from Abernathy Seed Company, 30-5-5 from Badlands
Lawn Care, and 20-10-20 from Crabgrass Corporation. How much of these fertilizers does he need to blend
to get the proportion of N-P-K he desires?
SKILL REVIEW
Below are skills that you will need in order to do well on Chapter 3. You are responsible for the information on
this sheet. If you have any questions you need to come see me in ac lab or before/after school.
Concept #1: Properties of Exponents
1. 53 51

4. x 2 y 3
7.

5
a2
a6
10. (10e4 x )3
 
2. x 5 x 3 x 2
3. a5
5.  2xy 
6. 32
8.
0
32 xy 5 z
24 x 2 y 2
11. 2e3 6e5
3
9. 24 8x
12.
24e8
4e5
Concept #2: Logarithmic and Exponential form
Write each log function in exponential form and write each exponent in logarithmic form.
13. log2 8 = 3
14. log1/ 4 4  1
15. 43 = 64
17. log27 3
18. Log5125
Concept #3: Evaluate
16. log232
Concept #4: Expanding/Condensing Logs
Expand the following logs
19. log3 4x
20. log
12
5
21. log x 2 y1/ 3
22. log
6 x2
y4
Condense the following logs
23. log47 – log410
24. 6logx + 4 log y
25. 5log2 +7logx -4logy
26. 6log2 – 4 logy
Chapter 3: Exponential & Logarithmic Functions
Essential Questions
What is an exponential function?
What is exponentiating?
What is a logarithmic function?
How are exponential and logarithmic functions similar to other functions that we have studied?
How are exponential and logarithmic functions different from other functions?
Learning Targets
Determine whether a function is exponential or logarithmic.
Graph all exponential and logarithmic functions.
Solve exponential and logarithmic equations.
Determine whether a set of data can best be modeled using a linear, polynomial, rational, exponential, or
logarithmic function.
Homework
Sections 3.1 - Exponential Functions & Graphs
Graph the function. Identify any asymptotes.
1. f ( x)   32 
x
2. f ( x )   32 
x
3. f ( x )   32 
x2
4. f ( x)   32 
x
2
For problems 5-8, match the function to the graph.
(a)
(b)
5. f ( x)  2 x2
(c)
6. f ( x)  2 x
(d)
7. f ( x)  2 x  4
8. f ( x)  2 x  1
10. f ( x)  2 x1
11. f ( x)  e x
12. f ( x)  2e0.5 x
14. s(t )  3e0.2t
15. g ( x)  1  e x
Sketch the graph of the function.
9. f ( x )   52 
x
13. f ( x)  4 x3  3
16.
Compound Interest
Complete the table for balance A using the appropriate compound interest formula.
P = $1000
r = 6%
t = 10 years
n
1
2
12
365
Continuous
A
17.
Radioactive Decay
Let Q represent a mass of Carbon 14 (14C), in grams, whose half-life is 5730 years. The quantity present
after t years is given by Q  10  12 
(a)
(b)
(c)
(d)
18.
t 5730
.
Determine the initial quantity (t = 0)
Determine the quantity present after 2000 years.
Graph the function Q over the interval from t = 0 to t = 10,000
When will this quantity of 14C be 0 grams? Explain your answer.
Population Growth
The population of a town increases according to the model P  2500e0.0293t , where t is the time in
years, with the year 2000 corresponding to t = 0 (e.g. This year is 2011).
(a) Graph the function for the years 2000 through 2025.
(b) Approximate the population in 2015 and 2025 using the graph.
(c) Verify your values in (b) using the model (i.e. evaluate the function at those times).
Section 3.2 - Logarithmic Functions & Graphs
Write the logarithmic equation in exponential form.
1. log3 81  4
1
2. log10 1000
 3
3. log16 8 
3
4
4. ln 4  1.386...
Write the exponential equation in logarithmic form.
5. 82  64
7. 103  0.001
6. 93 2  27
8. e x  4
Evaluate the function at the value of x without using a calculator.
9. f ( x)  log16 x at x 
10. f ( x)  log10 x at x  10
1
4
Solve the equation.
11. log5 5  x
12. log 2 21  x
13. log 3 43  x
Sketch the graph of each function. Make sure to show the x-intercept and vertical asymptote.
14. g ( x)  log 6 x
15. g ( x)  log 2 ( x)
17. y  log10 ( x  1)  4
18. f ( x)   log3 ( x  2)  4
16. g ( x)   log6 ( x  2)
For problems 19-22, match the function to the graph.
(a)
19. f ( x)  log3 x  2
(b)
(c)
20. f ( x)   log3 x
(d)
21. f ( x)   log3 ( x  2)
22. f ( x)  log3 (1  x)
Simplify using properties of natural logarithms.
23. ln e 2
27.
24.  lne
25. e ln1.8
26. 7 ln e0
 x 
 , if x > 750, approximates the length of a home
 x  750 
Home Mortgage: The model t  16.625ln 
mortgage of $150,000 at 6% in terms of monthly payments. In the model, t is the length of the
mortgage in years and x is the monthly payment in dollars.
(a) Use the model to approximate the length of this mortgage when the monthly payment is $897.72.
Approximate the total monthly payments over the term of this mortgage.
(b) Use the model to approximate the length of this mortgage when the monthly payment is $1659.24.
Approximate the total monthly payments for this option.
(c) What can you conclude from (a) and (b) regarding these payment options?
Section 3.3 - Logarithmic Properties
Rewrite the logarithm as a ratio using the change of base formula. Use both the common and natural logarithm
bases.
1. log3 x
2. log a 15
3. log y x
Use properties of logs to simplify the following expressions.

4. log 2 42  34

5. ln
 
6
e2
6. ln 5e 6
Use properties of logs to expand the following expressions.
7. log10
y
2
8. ln
xy
t
9. ln
x
10. log b
x2  1
x y4
z4
Use properties of logarithms to condense the following expressions.
11. 2ln8  5ln x
12. 3ln x  2ln y  4ln z
13. 2ln x  ln( x  1)  ln( x 1)
14.
1
2
ln( x  1)  2ln( x 1)  3ln x
Simplify to an exact value without a calculator.
15. log6 3 6
1
16. log5  125

17. log 4  16 
18. log 4 2  log 4 32
19. ln e 6  2ln e5
20. ln e 4.5
22.
21. ln 5 e3
Students participating in a psychology experiment attended several lectures and were given an exam.
Every month for the next year, the students were retested to see how much of the material they
remembered. The average scores for the group are given by the human memory model
f (t )  90 15log10  t  1
0  t  12 where t is the time in months from the first test.
(a) Graph the function over the given domain. You may use a calculator to help find points.
(b) Find the average score on the original exam (t = 0).
(c) Find the average score after 6 months. (d) Find the average score after 12 months.
(d) Find the time when the average score has dropped to 75.
Section 3.4 - Solving Exponential & Logarithmic Equations
Solve each equation. Leave each answer in exact form.
1. 7 x 
1
49
2. ln x  ln 2  0
3. e x  0
5. log x 25  2
1
6.    32
2
x
4. ln(3 x  5)  8
Solve each equation. Round each answer to three decimal places.
7. 4e 2 x  40
8. 14  3e x  11
10. log10 x 2  6
11. 4log10  x  6  11
9.
525
 275
1  e x
12. log3 x  log3 ( x  8)  2
13. log10 4 x2  log10  4  x   2


14. The demand equation for an iPad is given by p  5000 1 
4
4e
0.0002 x

 where x is the number of iPads

sold and p is the price per unit.
(a) Find x for p = $600.
(b) Find x for p = $800.
15. The yield V (in millions of cubic feet of timber per acre) for a forest at age t years is given by
V  6.7e 48.1/ t .
(a) Find the yield after 20 years.
(b) Find the time needed to obtain a yield of 1.3 million cubic feet.
Section 3.5 - Exponential and Logarithmic Models
For problems 1-6, match the function with its graph.
(a)
(b)
(d)
(e)
1. y  2e x / 4
4. y  3e ( x2)
7.
(c)
2
/5
(f)
2. y  6e x / 4
3. y  6  log10 ( x  2)
5. y  ln( x  1)
6. y 
4
1  e 2 x
Consider an initial investment of $20,000 earning interest at 10.5% compounded continuously.
(a) Find the amount of time needed to double the investment.
(b) Find the interest rate needed to double your investment in four years.
8. Find the exponential model y  aebx that fits the following points: (0,1) and (3,10)
9.
The number N of bacteria in a culture is given by the model N  250ekt , where t is the time (hours). If N =
280 and t = 10, estimate the time required for the population to double in size.
10. The amount Y of yeast in a culture is given by the model
Y
(a)
(b)
(c)
(d)
663
1  72e0.547 t
, 0  t  18 where t represents time (hours).
Make a table of values and graph this function over the given domain.
Use the model to predict the population for the 19th hour (e.g. t =19) and the 30th hour.
Find the limiting value of the population described by this model.
Explain why the population of yeast follows a logistic growth model instead of
an exponential decay model.
11. The Richter scale measures the magnitude R of an earthquake of intensity I using the model
R  log10
I
, where I0 = 1.
I0
(a) Find the intensity of the March 11, 2011 earthquake off the coast of Japan (R = 9.0).
(b) Find the intensity of the February 7, 1812 earthquake near New Madrid, Missouri (R  7.0).
(c) Find the magnitude of an earthquake with an intensity of 251, 200.
12. The level of sound β (in decibels) is related to sound intensity I using the model
  10log10 ( I / I 0 ) , where I0 = 10-12 watts per m2, which is roughly the faintest sound that can be heard by
the human ear.
(a) Calculate the level of sound (a.k.a. decibel level) of a quiet room (I = 10-10 watts per m2).
(b) Calculate the decibel level of a loud car horn (I = 10-3 watts per m2).
(c) An F-16 fighter jet has a sound level of about 90 decibels (dB) when landing, while the
newer F-35 jet lands with a sound level of 105 dB. Calculate the difference in sound intensity
represented by this 15 dB difference in sound level.
Chapter 3 Review
1. Determine the exponential function whose graph is shown in the figure.
y  aebx
Determine if the following are an exponential growth function or an exponential decay function.
2. y  e4 x  2
3. y  2(5x 3 )  1
Graph the function and analyze it for domain, range, and asymptotes.
5. y  3x  2  5
4. y  log( x  3)  2
Evaluate the logarithmic expression without using a calculator.
6. log 2 32
7. log3 81
8. log 3 10
9. ln
1
e7
Rewrite the equation in exponential form.
11. log 2 x  y
10. log3 x  5
Expand the following:
12. ln
xy

z
13. log 3
x y4

z4
Condense the following:
15. 3[ln( x  1)  2ln( x  1)]  2ln( x 2  1) 
14. 2ln8  5ln z 
Solve the equation.
16. 10 x  4
17. e x  0.25
18. 1.05x  3
19. ln x  5.4
20. log x  6
21. 3x3  5
22. 3log 2 x  1  7
23. 2log3 x  3  4
Solve the equation.
24.
50
 11
4  e2 x
25. log( x  2)  log( x  1)  4
26. ln(3x  5)  ln(2 x  1)  ln 4
27. Find the amount A accumulated after investing $450 for 3 years at an interest rate of 4.6% compounded
annually.
28. Find the amount A accumulated after investing $4800 for 17 years at an interest of rate 6.2% compouned
quarterly.
29. How long would it take for your investment to double if it is compounded continuously at 8.5% interest
rate?
30. If Jane invests $1500 in a savings account with a 6% interest rate compounded monthly, how long will it take
until Jane’s amount has a balance of $5200?
31. The sales S (in thousands of units) of a cleaning solution after x hundred dollars is spent on advertising are
given by S  10(1  ekx ) . When $500 is spent on advertising, 2,500 units are sold. Complete the model by
solving for k. Estimate the number of units that will be sold if advertising expenditures are raised to $700.
32. (3x3)3 is equivalent to:
b. 9x6
a. x
c. 9x9
d. 27x6
e. 27x9
33. In the real numbers, what is the solution of the equation 82x+1= 41-x?
a. -1/3
b. -1/4
c. -1/8
d. 0
e. 1/7
34. If log3 2 = p and log3 5 = q, which of the following expressions is equal to 10?
a. 3p+q
b. 3p + 3q
c. 9p+q
d. pq
e. p + q
35. Whenever x, y, and z are positive real numbers, which of the following expressions is equivalent to 2 log3 x +
½ log6 y – log3 z?
a.
log3 (
 x2 
  log 6
 z 
x2 y
)
z
d. log3  x  z   log 6
b. log3 
 y
 y
 z 
 y
 log 6  
2 
x 
2
c. log 3 
 y

2
e. 2 log 3  x  z   log 6 
MORE EQUATION PRACTICE:
1
log 3 x  2 log 3 2
2
1. log 4 ( x  2)  log 4 8
2. log5 (2 x  3)  log5 3
3.
4. 2log 4 x  log 4 9
5. 2log5 x  3log5 4
6. 3log 2 x   log 2 27
7. 3log 2 ( x  1)  log 2 4  5
8. 2log3 ( x  4)  log3 9  2
9. log x  log( x  15)  2
10. log 4 x  log 4 ( x  3)  1
11. ln( x  1)  ln x  2
12. 2 x  10
13. 3x  14
14. 8 x  1.2
15. 2 x  1.5
16. 5(23 x )  8
17. 0.3(40.2 x )  0.2
SKILL REVIEW
Below are skills that you will need in order to do well on Chapter 4. You are responsible for the information on
this sheet. If you have any questions you need to come see me in ac lab or before/after school.
Concept #1: Trig Functions in your calculator. Evaluate.
1. sin 36 =
2. cos 15 =
3. tan 67 =
4. tan θ = 1.254
5. cos θ = .8932
6. sin θ = .2431
Concept #2:SOHCAHTOA. Use trig functions to find the missing sides and/or angles
7.
8.
9.
10.
85
40
x
Concept #3: Special Right Triangles
Use special right triangle relationships to solve for all missing sides. NO DECIMALS!!
11.
12.
13. Find the diagonal of a square with a length of 5
14. Find the area of an equilateral triangle with a side length of 8
Chapter 4: Trigonometry
Essential Questions
What is trigonometry?
How are trigonometric functions similar to other functions we have studied? How are they different?
What is the Unit Circle? Why do we study the unit circle?
Why do certain strategies help you to verify trigonometric identities?
How are the solutions to trigonometric equations similar to other equations we have studied? How are they
different?
Learning Targets
Solve any 30-60-90 or 45-45-90 right triangle.
Define all six trigonometric functions.
Determine all of the values of the six trigonometric functions on the unit circle.
Determine the values of the six trigonometric functions for any point in the Cartesian coordinate plane.
Analyze and graph any trigonometric function given its equation and vice versa.
Determine whether a set of data can be properly modeled using a trigonometric function.
Section 4.3 - Right Triangle Trigonometry
1.
Find the exact value of the six trigonometric functions of the angle θ.
(a)
(b)
2.
Sketch a right triangle corresponding to the trigonometric function. Find the exact value of the other five
trigonometric functions.
(b) cos 
(a) cot   5
(c) csc  174
3
7
For problems 3-4, find the indicated trigonometric function values using the given function values.
3
1
, cos 60  : (a) tan 60 (b) sin30 (c) cos30 (d) cot 60
2
2
3.
sin 60 
4.
csc  3, sec 
5.
cos  
3 2
: (a) sin 
4
1
: (a) sec
4
(b) sin 
(d) sec  90   
(b) cos
(c) tan 
(c) cot 
(d) sin  90   
For problems 6-7, use trigonometric identities to transform one side of the equation into the other.
cot  sin  cos
6.
csc tan  sec
8.
Mr. Wu stands 65 meters from the base of the Jin Mao Building in Shanghai, China. He looks up at Mr.
Coulson, who on the 88th floor observation deck, and estimates that the angle of elevation from the street
to the top of the 88th floor is 80○. Calculate the approximate height of the building, and the distance
between Mr. Wu and Mr. Coulson at this point.
9.
In traveling across eastern Colorado you see the front range of the Rockies directly in front of you. For
some reason, you stop the car and measure the angle of elevation to the closest peak to be 3.5○. After you
drive due west another 13 miles, the urge again seizes you to stop and measure the angle of elevation to
this same peak. Now it is 9○. Using this information, find the approximate height of that mountain.
7.
Section 4.1 - Angles, Radians & Degrees
1.
Determine the quadrant in which each angle lies. Then sketch in standard position.
(a) 

12
(b) 257.5
(d) 460.25
(c) 2.25
2.
Determine two coterminal angles in (one positive, one negative) for each angle.
(a)
(b)
(c)
3.
Find (if possible) the complement and supplement of each angle.
(a)   61
4.
3
4
(c)  

2
(d)   183
Rewrite each angle in either degrees or radians (exact value, without calculator).
(a)   315
5.
(b)  
(b)   270
Find the angle in radians.
(a)
(c)   
7
6
(d)   4
(b)
6.
Find the length of the arc on a circle with a radius of 9 feet and a central angle of 60○.
7.
A car is moving at a rate of 40 miles per hour, and the diameter of its wheels is 2.5 feet.
(a) Find the linear speed of the tires in feet per minute.
(b) Find the number of revolutions per minute the wheels are rotating.
(c) Find the angular speed of the wheels in radians per minute.
Section 4.2 - Unit Circle
1.
Determine the exact values of the six trigonometric functions of the angle θ.
(a)
(b)
(e)

6
(e)   2
2.
Find the point on the unit circle that corresponds to the angle θ.
(a)  
3.
3
4
5
6
(c)  
11
6
(b)  
3
2
Find the value of the given trigonometric function.
(b) sin
 19 

 6 
9
4
(c) sin  
For the given trigonometric function, find the value of the indicated function.
(a) Given cos   
7.
4
3
(b)  
(a) cos7
6.
(c)   
Find the six trigonometric functions for the angle θ.
(a)  
5.
5
4
(b)  
Find the sine, cosine and tangent for the angle θ.
(a)  
4.

3
3
, find cos(  )
4
(b) Given sin(  ) 
3
, find csc( )
8
A bocce ball suspended from a Slinky bobs up and down, but because of friction the ball moves up and
down less with each cycle. This is called damped harmonic motion, and in this case the vertical position of
t
the ball y (in feet) is given by the function y (t )  14 e cos 6t , where t is the elapsed time (in seconds).
Find the position of the ball at the following points in time.
(b) t 
(a) t  0
(c) t 
1
4
1
2
Section 4.4 - Trig Functions for Any Circle
1.
Determine the exact values of the six trigonometric functions for the angle θ.
(a)
2.
(b)
Find the values for the six trigonometric functions of θ.
(a) cos   54 and θ lies in Quadrant II (b) csc  4 and cot   0
3.
Find the reference angle θ’ for the angle θ. Sketch both angles in standard position.
(a)   225
4.
(b)  
3
4
(c)   95
(d)   1.72
Find the indicated trigonometric value from the given function value and quadrant.
(a) Find sin  when cot   3 and θ lies in Quadrant II
(b) Find cot  when csc  2 and θ lies in Quadrant IV
(c) Find tan  when sec   94 and θ lies in Quadrant III
(d) Find csc when tan    54 and θ lies in Quadrant IV
5.
Find the two solutions for each trigonometric equation on the interval 0    2 .
(b) cot    3
(a) csc   2
6.
A company that produces water skis forecasts monthly sales over a two-year period using the following
model:
S  23.1  0.442t  4.3sin
t
where S is sales in thousands of units and t is time in months.
6
(a) If t = 1 represents January of 2011, estimate sales for this month.
(b) Using the same basis, estimate the sales for June 2011.
(c) Explain how this model accounts for the seasonal nature of water ski sales.
Section 4.5 - Sine and Cosine Functions
1.
Find the period and amplitude of each function.
(a) y  2cos3 x
2.
3.
x
3
3
x
cos
4
12
(c) y 
Describe how f(x) and g(x) differ (consider amplitudes, periods and shifts).
(a) f ( x)  sin 3x, g ( x)  sin( 3x)
(b) f ( x)  cos x, g ( x)   12 cos x
(c) f ( x)  cos x, g ( x)  cos( x   )
(d) f ( x)  cos 4 x, g ( x)  6  cos 4 x
Sketch the graphs of f(x) and g(x) on the same coordinate plane. Show one period.
(a) f ( x)  sin x, g ( x)  sin
4.
(b) y  3sin
x
3


(b) f ( x)   cos x, g ( x)   cos  x 
Sketch the graph of the function by hand. Show one period.
(a) y  4sin  x  2

2
x 
3
3
(c) y  4sin 


(b) y  6cos  x 


3

x
x  3
3
 3
(d) y  3cos 


2
5.
Mr. Coulson’s resting blood pressure P (measured in mm Hg) is modeled by
P  80  20cos
(a)
(b)
(c)
(d)
8
t
3
where t is the time (in seconds).
Graph the model.
Given that one period is equal to one heartbeat, find the time required for one heartbeat.
From the time per heartbeat, find Mr. Coulson’s resting pulse rate in heartbeats per minute.
Find Mr. Coulson’s maximum and minimum resting blood pressures.
Section 4.6 - Secant and Cosecant Functions
For problems 1-3, match the function with its graph. Find the period of the function.
(a)
1.
(b)
y  sec
x
2
(c)
2. y 
1
x
sec
2
2
3. y   csc x
For problems 4-9, sketch the graph of the function. Show one period.
4. y 
1
sec x
4
5. y  2csc  x
x 
 2
3 3
6. y  2sec 4 x  2
8. y  2sec  4 x   
7. y   csc 
9. y  2csc  2 x     1
Section 4.6 - Tangent and Cotangent Functions
For problems 1-3, match the function with its graph. Find the period of the function.
(a)
1. y  tan
(b)
x
2
2. y  tan 2 x
(c)
3. y  cot
x
2
For problems 4-9, sketch the graph of the function. Show one period.
4. y  cot x
8. y 
1
cot  x   
4
5. y  3tan 4 x
9. y 
6. y  3cot  x
7. y  
1
tan 2 x
2
1
x  
tan 
 
2
 4 4
Chapter 4 Review
1. For a right triangle, tan   3 . Find the value of the other five trig functions.
2. For a right triangle,
1
1
 . Find the value of the six trig functions.
csc 2
3. Name the quadrant that the angle x lies in given that sec x > 0, and sin x < 0.
4. Name the quadrant that the angle x lies in given that csc x > 0, and tan x < 0.
5. Suppose for an angle θ, cot θ = 7/24 and sec θ < 0. Find the exact value of sin θ.
6. Convert 75˚ to radian measure.
7. Convert -7π/4 to its exact degree measure and determine the quadrant of the terminal side of the angle.
8. Find the complement and supplement of π/5.
9. Find the reference angle for θ = -2π/3. Then find the exact values of the six trigonometric functions of
θ=2π/3.
10. If tan θ = 1/2 and sin θ < 0, find the quadrant of θ and the exact values of the remaining five trigonometric
functions of θ.
11. If sec θ = -5/4 and tan θ < 0, find the quadrant of θ and the exact values of the remaining five trigonometric
functions of θ.
12. Identify the phase shift (horizontal shift), amplitude, and period of y = 4sin(3x – π).
13. Find the amplitude and period of the sinusoidal graph given, then write an equation of the graph.
14. Given that sec x = 3, find cot x.
15. An escalator 152 feet in length rises to a platform and makes a 37 angle with the ground. Find the height
of the platform.
16. A man at the top of a ramp 30 feet in length looks down to the end of the ramp which rises to a loading
platform 3 feet off the ground. For a safe ramp, the tan  0.15 . Is this ramp safe?
17. cos 240˚ =
18. cot 13π/6 =
21. sec 330˚ =
22. csc 5π/3 =
19. sin 120˚ =
20. tan π/4 =
23. Use the fundamental identities to determine the simplified form of the expression.
a) cos θ csc θ =
b) tan θ cot θ =
c) sin θ cot θ =
24. Sketch one period of the graph of f ( x)   sin ( x   )  1 . Find the period, amplitude, and phase shift.
25. Sketch one period of the graph of f ( x)  2 tan ( x 
26. Sketch one period of the graph of f ( x)  sec ( x 

4

2
) . Find the period and phase shift.
)  2 . Find the period and phase shift.
27. Mr Coulson’s oven doesn’t bake cookies well, and he suspects the temperature controller is off. The
repairman suggests that when the oven is set at 350 F , the temperature should fluctuate no more than
10 F during a 7 minute cookie-baking cycle, and that it follows a sinusoidal path. What would the
equation for this temperature variation look like? Graph the expected equation over the course of the
hour that Mr. C bakes cookies. Sketch a second graph of possible test data that would confirm Mr. C’s
suspicion.
28. How can a right triangle have negative values for trigonometric functions?
29. What is the measure of the reference angle for an angle of 300˚?
a. 30˚
b. 60˚
c. 120˚
d. 150˚
e. 300˚
30. The 2 legs of a right triangle measures 37 inches and 45 inches, respectively. What is the cosine of the
triangle’s smallest interior angle?
a.
37
45
b.
45
37
c.
37
37  45
d.
37
37  45
2
2
e.
45
37  452
2
31. The figure below shows the path of a certain projectile launched from the ground at an angle of θ. The
horizontal range, r, of this projectile when launched from the ground at a speed of 20 meters per second is
modeled by r = 40 sin (2θ).
For this model, the angle measure θ that results in the greatest horizontal range, r, is 45˚ because:
a.
b.
c.
d.
e.
2 sinθ is greater than sin θ
sin 90˚ is as large as sine can get.
sin 45˚ is as large as sine can get.
sin 45˚ is about 0.707
sin 2θ is greater than sin θ.
32. The domain of the function y(x) = 3 cos(5x – 4) + 1 is all real numbers. Which of the following is the range of
the function y(x)?
a. 3  y ( x)  3
b. 4  y ( x)  3
c. 4  y ( x)  2
d. 2  y ( x)  4
e. All real numbers
Chapter 6: Law of Sines and Cosines
Essential Questions
Why do we have an ambiguous case?
How do you solve a triangle when it is not right?
How you determine which method to use (law of sines or law of cosines)?
Learning Targets:
Solve non-right triangles.
Determine how many triangles are formed in the ambiguous case.
Know when and how to solve the ambiguous case.
Section 6.1 – Law of Sines
Solve the triangle(s) or write none if a triangle cannot be formed.
1. C = 105˚, c = 20, B = 40˚
2. C = 135˚, c = 45, B=10˚
3. A = 60˚, a =9, c = 10
4. A = 24.3˚, C = 54.6˚, c = 2.68
5. A = 110˚, a = 125, b = 200
6. A = 76˚, a = 34, b = 21
7. A = 58˚, a = 4.5, b = 12.8
8. A = 130˚, a = 92, c = 30
Section 6.2 – Law of Cosines
Solve the triangle.
1. a = 90, b = 3, c = 11
2. C = 108˚, a = 10, b = 7
3. a = 45, b = 30, c = 72
4. a = 1.42, b = 0.75, c = 1.25
5. A = 50˚, b = 15, c = 30
6. The baseball player in center field is playing approximately 330 feet from the television camera that is behind
home plate. A batter hits a fly ball that goes to the wall 420 feet from the camera. The camera turns 6˚ to
follow the play. Approximate the distance the center fielder has to run to make the catch.
Chapter 6 Review
Solve the triangle(s) or write none if no triangle can be formed.
1. A = 62˚, a = 10, b = 12
2. A = 95˚, C = 35˚, c = 18
3. A = 54˚, a = 6, b = 7
4. A = 72˚, B = 35˚, c = 21
5. a = 13, c = 15, B = 25˚
6. a = 12, b = 18, c = 17
7. a =25, b = 42, c = 56
8. A = 60˚, b = 50, c = 48
8. In ABC below,  A measures 101˚,  B measures 25˚, and the length of BC is 165 meters. To the nearest
meter, what is the length of AC ?
a. 41
b. 54
c. 64
d. 71
e. 81
9. In the figure below, a radar screen shows 2 ships. Ship A is located at a distance of 20 nautical miles and
bearing 170˚, and Ship B is located at a distance of 30 nautical miles and bearing of 300˚. Which of the following
is an expression for the straight-line distance, in nautical miles, between the 2 ships?
a.
202  302  2(20)(30) cos 600
b.
202  302  2(20)(30) cos1300
c.
202  302  2(20)(30) cos1700
d.
202  302  2(20)(30) cos 3000
e.
202  302  2(20)(30) cos 4700
Final Review
1) log 1 
2) log 2 8 
5) log16 4 
6) log  6 
9) log 2 
1

25
1
7) log 36 
6
1

729
9
8) log 2 
3 4
3) log 5
4) log 9
1

8
Graph the following functions.
10) f ( x)   log ( x  1)  6
11) f ( x)  1  ln( x  4)
12) f ( x)  3  2 x  4  6
13) f ( x)  2  3x 1  1
14) Expand the expression.
log 2
3( x  2)
y
15) Condense the expression.
4
1
2 ln( x  1)  4 ln y 4   5ln( z )


3
16) Solve the equations algebraically.
A)
1025
5
8  e4 x
E) 2ln 4x  15
B) log x  log(8  5 x)  2
C) 24 x1  13  35
D) 3e 5 x  132
F) log( x  2)  log x  log( x  5)
17) You will need $1,000,000 to retire at 63, and live comfortably. If you expected a 7% return on your
investments, starting at the age of 18, how much money would you have to invest at age 18 to reach your goal?
Assume interest is compounded continuously.
18) What would earn you more money, an account with 8% interest compounded continuously, or an account
with 8.2% interest compounded monthly?
19) Name the quadrant that the angle x lies in given that sec x < 0, and sin x > 0.
20) Name the quadrant that the angle x lies in given that csc x > 0, and tan x > 0.
21) Find cos 625 
22) Suppose for an angle θ, cot θ = 4/3 and sec θ < 0. Find the exact value of sin θ.
23) Convert 275˚ to radian measure.
24) Convert 
7
to its exact degree measure and determine the quadrant of the terminal side of the angle.
6
25) Find the complement and supplement of

.
7


26) Identify the phase shift (horizontal shift), amplitude, and period of f ( x)  3sin  5 x 
27) Find the reference angle for θ = 

6

 1 .
2
. Then find the exact values of the six trig functions of θ = 

6
.
28) If sec θ = -9/4 and tan θ > 0, find the quadrant of θ and the exact values of the remaining five trigonometric
functions of θ.


29) Find the amplitude, period, and phase shift of the function f ( x)  1  3cos  2 x 

 . Then sketch a graph
2
of the equation with one full period.
30) Find the period and phase shift of the function f ( x)  2 tan  4 x     1 . Then sketch a graph of the
equation with one full period.


31) Find the period phase shift of the function f ( x)  4 csc   x 

  1 . Then sketch a graph of the
2
equation with one full period.
32) Find the amplitude and period of the sinusoidal graph given, then write an equation of the graph as BOTH a
sine and cosine function.
33) Given that sec x = 5, find cot x.
No Calculator!
34) cos 210˚ =
35) cot 15π/6 =
Use the fundamental identities to determine the simplified form of the expression.
37)
s in 
cot  sec
cos
36)
cos θ sec θ
38)
sin x cos x tan x cot x sec x csc x
39) A ramp rises to a platform that is 10 ft off of the ground. If the ramp itself is 15 ft long, what is the angle of
the incline of the ramp?
40) A ramp rises to a platform that is 20 ft off of the ground. The angle of elevation of said ramp is 5°. How
long is the ramp horizontally along the ground?
41) Explain, in words, how you would solve a non-right triangle, using the Law of Sines and Law of Cosines. Be
specific.
OVER BREAK:
To Prepare for next semester answer the following questions.
Concept #1: Multiplying/Dividing/Adding/Subtracting Fractions
Evaluate. Leave answers in fraction form.
1.
3 7

4 4
2. 
3
5
2
3
3.
12 3

5 25
4.
1
2
6
5.
2 5
7 2
6.
8
( 4)
9
7.
3 1

8 2
8.
3
3
7
9. 6 
9
2