Honors Precalculus Semester 1 Review
Complete on a separate sheet of paper. Show all work. Use your calculator ONLY on problems that require it.
Doing all these exercises will take several hours or more. Spread out the effort over a few days for maximum benefit.
Doing all of these exercises is not required. This is a suggested assignment for the students who are committed to
preparing to be successful. If there is a particular area where you struggle more, then exercises found in the homework
sets on those sections in the actual chapters should provide additional opportunities for review. You will get points for
the effort made on the review exercises. Study hard!
Can you read a graph?
1. Given the graph of y= f(x) pictured at the right, find each of the
following.
a. The domain of f .
b. f (0)
c. The value(s) of x for which f(x) = -15 .
d. The x-intercepts of the graph.
e. The values of x where is f increasing.
f. The number of times the line y = -10 crosses the graph.
g. The graph appears to be two functions. State the type of each
function and domain of each. Degree for each function?
2. Answer the following questions about the four graphs shown below. Some questions can have more
than one answer.
a. Which graphs, if any, are not graphs of a function?
b. Which graphs are defined for all x?
c. Which graphs are graphs of a constant function?
d. Which graphs have asymptotes?
e. Which graphs look like the graph of y = -2?
f. Which graphs are odd? Are even? Are neither?
3. Kenny, who got his driver’s license two weeks ago, got permission to
borrow his Dad’s car for the day. His plan was to drive around for a
while and then meet his friends at 3 pm. The graph you see at the
right tells you what Kenny’s day with his Dad’s car was like. The x-axis
shows the time of day with 0 denoting 8 am, and the y-axis shows
you how many miles Kenny was from his home.
Use this graph to answer the following questions about Kenny’s day.
a. How fast, on the average, was Kenny driving between 9 and
10am?
b. What was Kenny doing between 10:00 and 10:15 am?
c. What does the decreasing part of the graph indicate?
d. Did Kenny take his friends for a ride in his Dad’s car at 3 pm? Explain.
Do you recognize types of functions and can you find equations for them?
4. Describe the type of function (and how you know), the general equation, whether the function is
increasing or decreasing, and its concavity.
Do you understand functional notation, domain, range, odd or even symmetry, end behavior, etc.?
5. Scientists have found that the number of times a cricket chirps per minute is a function of the outdoor
10
temperature. In particular, they have found that the function f(x) = 3 (x-40) describes this relationship
when the temperature is measured in degrees Fahrenheit.
a. What does x denote: chirps per minute or temperature?
b. Without using any “math lingo”, describes in words the meaning of each of the following. Your
description should fit the context of the problem and should use words like “chirps”,
“temperature”, “hotter”, etc. Your description should make sense to a 5 year old child.
i. f(2x)
ii. 2f(x)
iii. f(x+ 5)
iv. f(x)+5
2
6. If f(x) = 3x + 4 and g(x) = x – 5x – 6 ,
a. sketch the graph of y= f(x) and g(x) .
b. find the x-intercept of y= f(x).
c. find the y-intercept of y= g(x).
d. find, and simplify, f (-4) and g(-4) .
e. find, and simplify, 3f(x) and 2g(x) when x = 3.
f. find x when f(x) =12 and g(x) = 0.
g. find, and simplify, f(x+ 5) and g(x – 2).
h. find, and simplify,
𝑓(𝑥+ℎ)− 𝑓(𝑥)
ℎ
where h ≠ 0 .
2
7. What is the domain of f ( x)  4  x ?
8. Write a function that expresses the area A of a circle as a function of the diameter D.
9. Let f ( x)  x  5 . Write an equation for a function g ( x ) whose graph can be obtained from the
function f ( x) by shifting the graph horizontally 4 units to the left and then vertically up 5 units.
3
10. Let g (t )  t  t and f ( x)  2  x . Find the value of g ( f (3)  1) .
2
11. Let f ( x)  x  2 and g ( x)  x . Find the value of ( f g )(2) .
2
12. Let f ( x)  4 x  1 and g ( x)  x  2 . Write the function f ( g ( x)) .
2 x
2
1
c. f ( x)  4  x
13. Find f ( x) for: a. f ( x)  5 x  2 b. f ( x) 
2x  3
14. This problem involves the composite function f(g(x)), where
f(x) = 0.5x + 1.5
for 1 ≤ x ≤ 5
g(x) = 8 – x
for 2 ≤ x ≤ 4
a. Explain why f(g(2)) is undefined even though both f(2) and
g(2) are defined.
b. Plot y = f(g(x)) on your calculator. Sketch the result on the
given figure.
c. From the graph, give the domain of function f ˚ g.
d. Find the domain and range of f ˚ g algebraically.
15. Use the piece-wise function at the right .
a. Find an equation for the piece-wise function, or explain
why you can’t.
b. Sketch the inverse of the function on the same set of axes.
Do you understand linear and quadratic functions?
16. Find the slope of the line containing the points (6, 4) and ( 8, 2) .
17. If the slope of one line is 3 and the slope of a second line is 3, what can be said about the lines?
18. What are the slope and y-intercept of the line 4 x  3 y  7 .
19. Find a number such that the line passing through the points ( k , 3) and (5, k  3) has a slope of 2.
20. Find an equation of the line with slope 2 that contains the point (4, 2) .
21. Find an equation of the line that passes through the points (5,3) and (2,8) .
22. Write an equation of the line parallel to y  3 x  4 with a y-intercept of -1.
23. Find an equation of the line perpendicular to 4 y  5 x  2 and containing the point (2, 5) .
24. Find an equation of the tangent line to the circle x 2  y 2  25 at the point (3, 4) .
25. Use your graphing calculator to determine whether ( x  2)4  x 4  16 . Give the viewing window you
used.
26. The height of a projectile as a function of time is given by the following equation:
ft.
m
g  32 2
g  9.8 2
2
1
s or
s v0  initial upward velocity and h0  initial
h(t )   gt  v0t  h0 , where:
2
height
a. How long does it take for an object to reach the ground if it is dropped from the top of a 1000
ft. building?
ft.
b. If a ball is thrown from an initial height of 6 ft., with an initial upward velocity of 20 , what
s
maximum height will it reach, and how long will it take to hit the ground?
c. When will the ball reach a height of 8 ft?
Do you understand polynomial functions?
27. What is the domain of the function f ( x)  3x5  x 4  3x ?
28. Graph a polynomial function by finding the zeros of a polynomial function.
a. By graphing,
f(x)= x4 −11x3 +34x2 −6x −72.
g(x)= x3 −x2 −7x +15
Explain how you know that two of the zeros of g(x) are complex numbers.
b. By synthetic substitution, e.g.,
h(x)= x3 −x2 −7x + 15
k(x) = x4 − 8x3 + 18x2 − 27.
Find the complex zeros using the quadratic formula.
4
3
2
29. Find the remainder when x  3x  x  5 is divided by x  1 .
3
2
30. Let f ( x)  x  2 x  x  6 .
a. List the possible rational zeros.
b. Find all rational roots
Sketch a graph of the function. Be sure to accurately indicate zeros and the y-intercept.
31. Find the particular equation of a polynomial function.
a. From points: a cubic function containing the points (2, 7), (3, 34), (4, 91), and (5, 190).
b. From zeros: a cubic function with leading coefficient 1 and zeros −5, 3, and 6. Write the
equation in factored form and then multiply and simplify. Confirm by graphing.
c. Using the Rational Root Theorem: list all possible rational zeros for f(x) = 4x3 - 7x - 3.
32. Solve equations involving polynomials.
a. x4 + x3 – 3x2 – x + 2 = 0
b. log 2 (x3 −3x2 −8x +12) + log 2x = 4
33. Find the domain of each of the following functions. Sketch the function without your calculator.
a. 𝑓(𝑥) = |𝑥 4 − 8𝑥 3 − 8𝑥 + 15|
b. 𝑔(𝑥) = √4 − 𝑥
34. A rectangular package to be sent by a delivery service can have a maximum combined length and girth
(perimeter of a cross section) of 120 inches.
a. Write an expression for the combined length and girth of the box. Use
the fact that this combined expression can be no more than 120
inches to write an expression of the height, y, in terms of the length
of the square bottom, x.
b. Show that the volume of the package is V(x) = 4x2(30 – x).
c. Sketch a graph of the function, V, and estimate the value of x for which the volume is
maximum. State the dimensions of the box that yields the maximum volume.
Do you understand rational functions and equation solving?
x 2  3x  4
35. Find the domain of the rational function and graph it. f ( x)  2
x  x6
2
x  x6
36. Let f ( x)  3
.
2 x  8 x 2  8 x  32
a. Sketch a complete graph of f(x). Describe the domain and range.
b. What are the vertical asymptotes? Removable discontinuities? Intercepts?
c. What is the end behavior asymptote?
37. Write the rational function for the graph below.
Solve the following equations and inequalities using appropriate methods.
Honors Precalculus Semester 1 Review
Complete on a separate sheet of paper. Show all work. Use your calculator ONLY on problems that require it.
2x  3  x  3
2x
x
2


39. 2
x  6x  8 x  4 x  2
38.
40. 4 2 x  1  12  0
2x  5
6
3
42. x 4  x 2  12  0 .
x3
43.
1
x2
41. 3 
Do you understand triangle trigonometry?
44. A UFO is first sited at a point P1 due East from an observer at an angle of 25˚
from the ground and at an altitude (vertical distance above the ground) of
100m. The UFO is next sighted at a point P2 due East at an angle of 50˚ and an
altitude of 500m. How far did the UFO travel from P1 to P2? (hint: what is the
distance?)
Do you understand trigonometric functions and the unit circle?
45. Sketch an angle of 300˚ in standard position.
a. Mark the reference angle on the diagram with red pen.
b. State the measure of the reference angle.
c. Write the sine and cosine ratios of 300˚ exactly.
d. Name two co-terminal angles to 300˚, one positive and one negative.
46. Name the reference angle for 2345˚
1
47. Show that
does not equal sin–1 0.5˚.
sin0.5
48. Draw the functions: y = cos θ, y = sin θ and y =tan θ. Include the scale and key points.
49. Draw and label all important points and angles (radians and degrees) of the unit circle.
50. If tan  is not defined and 0    2 , what is  ?
51. Find the exact value without a calculator.
7



 8 
a. sin
b. tan cos  cos
c. cot   
4
6
3
2
 3 
52. If (-2, 5) is a point on the terminal side of angle  , find the exact value of each of the six trigonometric
function of  .
53. Determine the quadrant in which θ lies if cos  0 and cot   0 .
1
54. If cos   and tan   0 , find the exact value of each of the remaining trigonometric functions of  .
3
55. Sketch two cycles. Clearly show asymptotes, vertices and inflection points. Identify the vertical
dilation or amplitude as appropriate, sinusoidal axes, period, interval lengths and phase shift.
Compare/contrast the three graphs.
a. y = 3 + 5 cos 15(𝜃 + 23˚)
b. y = 3 + 5 tan 15(𝜃 + 23˚)
c. y = 3 + 5 sec 15(𝜃 + 23˚)
56. Find the equation for this sinusoid, in both degree and radian forms
3
and, 0 ≤  ≤ , then  = ______ or ________ and tan  = __________
2
58. A radial arm saw has a circular cutting blade with a diameter of 10 inches. It spins at 2000 rpm. If
there are 12 cutting teeth per inch on the cutting blade, how many teeth cross the cutting surface each
second?
57. If sin  