PRE-CALCULUS
Semester 1 Review
Chapters 13 and 1 thru 5
Lesson 13.1.1 Circles
1.
2.
Find the equation of the circle(s) described below.
a.
C  (3, 6), r  12
b.
C  (2, 7), r  5 3
c.
C  (4, 2) and goes through (3, 1)
d.
C  (5, 1) and goes through (–2, 3)
e.
Diameter with endpoints (–5, –7) and (7, 7)
f.
Diameter with endpoints (2, 0) and (5, –3)
g.
C  (3,1) and tangent to the line x  8
h.
C  (8, 9) and tangent to the line y  1
Find the center and radius of the given circle.
a.
(x  2)2  (y  8)2  4
b.
(y  1)2  (x  3)2  48
c.
x 2  y2  16x  12y  88
d.
x 2  y2  12x  6y  53
3.
A circle is centered at (4, 8) and has an area of 294 . Find the equation of the circle.
4.
A circle is centered at (3, –7) and tangent to the line y  x  4 . Find the equation of the circle.
5.
A circle containing the point (8, 1) is tangent to the y-axis at the point (0, –3). Find the equation of the circle.
Lesson 13.1.2 Ellipses
1.
For the given ellipse(s), find the center, the lengths of the major and minor axes, the coordinates of the foci, and
sketch the graph.
a.
(x3)2
16
c.
(y2)2
25
b.
(x4)2
8
9x 2  16y2  144
d.
12x 2  9y2  108
e.
4x 2  9y2  16x  20  0
f.
x 2  6y2  4x  60y  136  0
g.
12x 2  5y2  24x  30y  39  0
h.
1
36

1

(x2)2
4
1
1 y2  1 x  9 y  153  1
x 2  25
2
25
50
2.
Find the equation of the ellipse with foci at (3, 7) and (3, –1) and a minor radius of 3.
3.
Find the equation of the ellipse with vertices at (5, 2) and (–7, 2) and focal point (4, 2).
4.
Find the equation of the ellipse that has a center at (–2, –5), a focal point of (2, –5), and a major axis of length 7.
5.
An ellipse is tangent to the x- and y-axes. The length of the major axis is 10 and a focal point is (8, –4). Find the
equation of the ellipse.
Lesson 13.1.3 Hyperbolas
1.
For the given hyperbola(s), find the center, the equations of the asymptotes, the coordinates of the foci, and sketch
the graph.
b.
(y1)2
36
4x 2  8x  12  4y2
d.
4x 2  y2  8x  6y  21  0
e.
x 2  4y2  2x  16y  51
f.
y2  4x 2  2y  16x  19
g.
2y2  4x 2  12y  8x  6  0
h.
x 2  16y2  4x  160y  400  0
a.
(x2)2
9
c.

(y2)2
25
1

(x3)2
16
1
2.
Find the equation of the hyperbola opening vertically, with asymptotes of y  2x and a vertical major axis of
length 8.
3.
Find the equation of the hyperbola with vertices at (–3, 4) and (–3, 0) and focal points at
(–3, 6) and (–3, –2).
4.
Find the equation of the hyperbola with focal points at (–1, 0) and (–1, 4) on the minor axis which is of length 2.
5.
Find the equation of the hyperbola with focal points at (–4, –4) and (4, –4) on the minor axis which is of length 6.
Lesson 13.1.4 Parabolas
1.
For the given parabola(s), find the vertex, focus, directrix, and sketch the graph.
a.
20x  y2
b.
1 x2
y   16
c.
12(y  1)  (x  3)2
d.
16(y  3)  (x  5)2
e.
y  (x  5)2  4
f.
x  4(y  4)2  2
g.
y  x 2  2x  3
h.
x 2  4x  8y  36  0
2.
Find the equation of the parabola with the focus at (7, –1) and directrix of x  5 .
3.
Find the equation of the parabola with the vertex at (6, –2) and directrix of y  9 .
4.
Find the equation of the parabola with the vertex at (–2, –4) and the focus at (3, –4).
5.
Find the equation of the parabola with the vertex at (–4, 3), containing the point (–6, 11).
Lesson 13.1.1 Answers
1.
2.
a.
(x  3)2  (y  6)2  144
b. (x  2)2  (y  7)2  75
c.
(x  4)2  (y  2)2  58
d. (x  5)2  (y  1)2  65
e.
(x  1)2  y2  85
f.
g.
(x  3)2  (y  1)2  25
h. (x  8)2  (y  9)2  64
a.
C  (2, 8), r  2
b. C  (3, 1), r  4 3
c.
C  (8, 6), r  2 3
d. C  (6, 3), r  7 2
3.
(x  4)2  (y  8)2  294
4.
(x  3)2  (y  7)2  18
5.
(x  5)2  (y  3)2  25
x  27   y  23   92
2
2
Lesson 13.1.2 Answers
y
y
1.
a.
b. C  (4, 2)
C  (3, 2)
x
M  10, m  8
F  (3, 5), (3, 1)
y
c.
C  (0, 0)
M  8, m  6
F  ( 7, 0), ( 7, 0)
x
M  4 2, m  4
F  (6, 2), (2, 2)
y
d. C  (0, 0)
x
x
M  4 3, m  6
F  (0, 3), (0,  3)
y
x
e.
C  (2, 0)
M  6, m  4
C  (2, 5)
f.
y
M  6 2, m  2 3
x
F  (2  5, 0)
g.
F  (2  15, 5)
y
y
C  (1, 3)
C  (9, 4.5)
M  12, m  10
h.
x
M  4 3, m  2 5
x
F  (9  11, 4.5)
F  (1, 3  7 )
*Graph is scaled by 3’s.
(x3)2
9
2.

(y3)2
25
3.
(x1)2
36

(y2)2
11
1
4.
(x2)2
49

(y5)2
33
1
5.
(x5)2
25

(y4)2
16
1
1
Lesson 13.1.3 Answers
1.
a.
C  (2, 2)
y
b.
y   53 (x  2)  2
C  (3, 1)
F  (8, 2), (4, 2)
*Graph is scaled by 2’s.
y
y
C  (1, 0)
y  (x  1)
F  (1  2 2, 0)
x
F  (3, 1  2 13)
*Graph is scaled by 2’s.
c.
y
y   23 (x  3)  1
x
x
d.
C  (1, 3)
y  2(x  1)  3
F  (1  2 5, 3)
x
C  (1, 2)
e.
f.
y
y   12 (x  1)  2
x
C  (2, 1)
y  2(x  2)  1
y
x
F  (2, 1  5 )
F  (1  3 5, 2)
*Graph is scaled by 2’s.
y
C  (1, 3)
g.
h.
x
y   2(x  1)  3
y2
16
3.
(y2)2
4

(x3)2
12
1
4.
(y2)2
1

(x1)2
3
1
5.
x2
9


x2
4
y
y   14 (x  2)  5

F  (1, 3  6 )
2.
C  (2, 5)
F  2 
17
2
,5
x

1
(y4)2
7
1
Lesson 13.1.4 Answers
y
y
1.
a.
V  (0, 0)
F  (5, 0)
D : x  5
*Graph is scaled by 2’s.
b.
x
V  (0, 0)
F  (0, 4)
D: y 4
*Graph is scaled by 2’s.
x
y
y
C  (3, 1)
c.
d.
F  (3, 4)
D : y  2
x
*Graph is scaled by 2’s.
V  (5, 3)
F  (5, 7)
D: y 1
x
*Graph is scaled by 2’s.
y
y
V  (5, 4)
e.

F  5, 4
1
4
x

f.
D : y  3 43
*Graph is scaled by 2’s.
V  (2, 4)
F  (3, 4)
D : x  1
x
*Graph is scaled by 2’s.
y
y
V  (1, 2)
g.

F  1, 2
1
4

D : y  1 43
h.
x
V  (2, 4)
F  (2, 8)
D: y0
*Graph is scaled by 2’s.
2.
x
1
4
(y  1)2  6
3.
y
1
28
(x  6)2  2
4.
x
1
20
(y  4)2  2
5.
y  2(x  4)2  3
x
PRE-CALCULUS
Semester 1 Review
Chapters 13 and 1 thru 5
Lesson 1.1.1 Investigating Linear Data
1.
At the beginning of the year, the second-period PE class members at South High were assess for weight (pounds)
and height (inches). Measurements of 10 boys in the class are listed in the following table.
Height
63
69
73
64
72
2.
Weight
130
155
195
150
185
Height
71
65
70
62
67
Weight
150
130
160
120
155
a.
Find a linear function for the above data. Let height be represented on the x-axis and weight on the y-axis.
Show all of your work clearly.
b.
What do the slope and y-intercept represent? Do you think your model is acceptable for people of all heights?
Explain.
Tasty Mornings is trying a variety of packaging dimensions for their cereal. They wish to predict the net weight of
the product based on the amount of cardboard used for the package. Below is a list of current packages with their
corresponding weights.
Packaging
(in2)
47
138
100
111
Weight
(grams)
28
850
283
425
Packaging
(in2)
125
69
88
Weight
(grams)
566
85
198
Find a linear function for the above data. Let packaging be represented on the x-axis and weight on the y-axis.
Show all of your work clearly.
3.
Find a linear equation to approximate the following set of data points. Show all of your work clearly.
(1, 3), (3, 9), (2, 5), (7, 8), (9, 10), (4, 7), (7, 11), (10, 13), (8, 15)
4.
Find a linear equation to approximate the following set of data points. Show all of your work clearly.
(1, 2), (2, 3), (3, 1), (3, 3), (5, 7), (6, 8), (8, 6), (9, 9)
5.
Find a linear equation to approximate the following set of data points. Show all of your work clearly.
(0, 2), (2, 2), (3, 4), (4, 7), (5, 7), (6, 9), (8, 9)
Lesson 1.1.2 Transformations of Parent Graphs
1.
For the parent graph f (x)  x 2 , sketch the original function following transformations on the same set of axes.
a.
2.
For the parent graph f (x) 
a.
3.
 f (x  1)
b.
f (x)  3
2 f (x)
x , sketch the original function following transformations on the same set of axes.
b.
3 f (x)  2
2 f (x  3)
b.
1
2
c.
f (x  1)  5
f (x  2)  4
c.
2
3
f (x  2)  2
Given the following function, state the equation of the parent graph, sketch the graph, and describe the
transformation completely.
f (x) 
5.
c.
For the parent graph f (x)  x , sketch the original function following transformations on the same set of axes.
a.
4.
f (x  2)
1
x5
3
Given the following function, state the equation of the parent graph, sketch the graph, and describe the
transformation completely.
f (x)  3(x  1)3  4
Lesson 1.1.3 Basic Inverses and Function Operations
1.
Given f (x)  x 2  8x and g(x)  3x  1 , find and simplify the following function operations.
a.
2.
2 f (x)  g(x)
b.
Given f (x)  2x 2  1 and g(x) 
a.
f (x  2)
f (x)  g(x)
c.
g( f (x))
2x  3 , find and simplify the following function operations.
b.
f (g(x))
c.
g( f (x))
3.
If f (x)  3x  4 and g(x)  x 2  3x  1 , find g( f (x)) and simplify completely.
4.
Find the inverse of g(x) 
5.
Find the inverse of f (x)  4(2x  3)1/3 and simplify completely.
2x  1 .
Lesson 1.1.4 Transformations of Non-Parent Graphs
1.
Given the graph of m(x), sketch the following transformations.
y
m(x)
a.
2m(x)
x
b.
m(x  2)  1
c.
m(x  1)  3
2. Given the graph of h(x), sketch the following transformations.
a.
h(x  3)  2
b.
1
2
y
h(x)  1
h(x)
x
c.
h(x)  4
3. Given the graph of f(x) at right, write an expression for each transformation
graph below in terms of f(x).
y
f(x)
x
a.
b.
y
y
x
x
4. Given the graph of f(x) at right, write an expression for each transformation
graph below in terms of f(x).
y
f(x)
x
a.
b.
y
y
x
x
Section 1.2 Point-Slope Form of a Line
1.
Find the equation of the line that is perpendicular to y  23 x  5 and passes through the point (8, 4) .
2.
Given two points, A(2,11) and B(6, 5) , complete the following problems.
a.
Find the slope of AB .
b.
Find the equation of the line parallel to AB , passing thought the point (100, 0).
c.
Find the equation of the line perpendicular to AB , passing thought the point (1, 3) .
d.
Find the equation of the perpendicular bisector of AB .
3.
4.
Find the point-slope form of the line that has the following characteristics.
a.
A slope of 56 and passes through the point (1.6, –3).
b.
Passes through the points (8.1, 1.2) and (–7.4, 19.8).
c.
Is perpendicular to y  45 x  7 and passes through the point (7, –3).
Which of the points, A(4, 4) or B(5, 3) , is closer to the point C(–1, –3) ?
5. Write the equation of the line that passes through the midpoint of, and is perpendicular to, a segment with endpoints
(–3, 4) and (4, –1).
6.
Find the distance between the point (–2, 4) and the midpoint of the segment with endpoints L(3, –5) and N(–11, 1).
Lesson 1.3.1 Law of Sines
1.
Given GEO , where G  32, E  81, and GE  12 feet.
a.
Draw a diagram roughly to scale.
b.
Solve the triangle completely.
c.
Calculate the area of GEO .
2.
Given ALG , where A  47º , G  53º , and AG  8 cm.
a.
Draw a diagram roughly to scale.
b.
Solve the triangle completely.
c.
Calculate the area of ALG .
3.
Marsha and Edwin want to know how far they will have to swim to get across a small lake. They have a 100-foot
rope that they place at the waters edge (in a straight line), then standing at each end, they use a compass to measure
the angle between the rope and a tree on the other side of the lake. Marsha’s angle is 67º. Edwin’s angle is 81º.
How far will they have to swim to get to the other side of the lake?
4.
Solve for x in the given triangle if the area is 50 cm2.
11 cm
26º
x
5.
Find the length of the missing leg and hypotenuse of the special right triangles below.
a.
b.
30
5 2
6
45


Lesson 1.3.2 Law of Cosines
B
1.
Solve the triangle at right.
7
5
A
3
C
N
22
2.
Solve the triangle at right.
I
61º
37
T
3.
Find the area of the triangle below. Show all of your steps.
13 cm
8 cm
15 cm
4.
Find the area of the triangle below. Show all of your steps.
10 m
12 m
19 m
5.
Two airplanes leave the airport at the same time. The angle between their paths (assume they fly in a straight line)
is 79º. If one plane is traveling at 300 mph and the other plane is traveling at 375 mph, how far apart are they after
1 hour? 2 hours?
Lesson 1.4.1 Radian Measure
1.
Convert the following angle measures from degrees to radians or radians to degrees.
a.
2.
75º
b.
7
12
radians
Convert the following angle measures from degrees to radians or radians to degrees.
a.
3 radians
b.
30 º
3.
Convert the following angle measures from degrees to radians or radians to degrees.
a.

3
5
radians
b.
250º
4.
A wheel is spinning at 430 revolutions per minute. How many radians per second is that?
5.
Mike notices that it takes him 5 seconds to ride his bike 100 feet. The wheels on his bike have a diameter of 26
inches.
a.
How many revolutions per second are the wheels making?
b.
How many radians per second are the wheels spinning at?
Lesson 1.4.2 Common Angles in the Unit Circle
1.
Find the angle between 0 and 2 that is coterminal with the given angle. Draw a picture of that angle in the unit
circle.
a.
2.
 3
b.
9
4
Find the angle between 0 and 2 that is coterminal with the given angle. Draw a picture of that angle in the unit
circle.
a.
15 
6
b.

4
3
3.
Find both a positive and a negative angle that are coterminal with 34 .
4.
Find both a positive angle and a negative angle that are coterminal with  32 .
5.
Find both a positive angle and a negative angle that are coterminal with 176 .
Lesson 1.1.1 Answers
1.
a.
Answers will vary. Using the Linear Regression function on a graphing calculator gives an answer of
y  5.25x  201 . Another possibility using the points (72, 185) and (63, 130) is y 
b.
55
9
x  253 .
The slope represents a person’s weight/inch. The y-intercept represents the weight of a person who is 0
inches tall. This indicated that this is not a good model for people who are short, like young children.
2.
a.
Answers will vary. Using the Linear Regression function on a graphing calculator gives an answer of
x  507 23 .
y  8.66x  491.26 . Another possibility using the points (69, 85) and (125, 566) is y  7 19
66
3.
Answers will vary. Using the Linear Regression function on a graphing calculator gives an answer of
.
y  0.98x  3.47 . Another possibility using the points (2, 5) and (7, 11) is y  65 x  13
5
4.
Answers will vary. Using the Linear Regression function on a graphing calculator gives an answer of
y  0.89x  0.75 . Another possibility using the points (2, 3) and (9, 9) is y  67 x  97 .
5.
Answers will vary. Using the Linear Regression function on a graphing calculator gives an answer of
y  1.07x  1.42 . Another possibility using the points (0, 2) and (5, 7) is y  1x  2 .
Lesson 1.1.2 Answers
1.
y
a.
c.
See graph at right.
x
2.
a.
Right 2 units.
b.
Down 3 units.
c.
Vertical stretch by a factor of 2.
b.
See graph at right.
y
a.
Vertical flip, up one unit.
b.
Vertical stretch by a factor of 3, down 2 units.
c.
Left 2 units, down 4 units.
b.
x
a.
c.
3.
See graph at right.
y
c.
a.
Vertical stretch by a factor of 2, vertical flip, right 3 units.
b.
Vertical compression by a factor of 2 (or stretch by a factor of 12
), left 1 unit, down 5 units.
x
b.
c.
Vertical stretch by a factor of 23 , right 2 units, up 2 units.
a.
4.
See graph below.
5.
See graph below.
Parent graph: y  1x
Parent graph: y  x 3
Right 5 units, up 3 units.
Vertical stretch by a factor of 3,
vertical flip, right 1 unit, down
4 units.
y
y
x
x
Lesson 1.1.3 Answers
1.
a.
2x 2  19x  1
b.
3x 3  25x 2  8x
c.
2.
a.
2x 2  8x  9
b.
4x  5
c.
3.
9x 2  15x  3
4.
g1 (x) 
5.
x  3
f 1(x)  128
2
x 2 1
2
3
3x 2  24x  1
4 x2  1
Lesson 1.1.4 Answers
1.
See graph at right.
y
c.
a.
Vertical stretch by a factor of 2, vertical flip.
b.
x
b.
Left 2 units, down 1 unit.
a.
c.
2.
Right 1 unit, up three units.
See graph at right.
y
c.
a.
Left 3 units, down 2 units.
b.
Vertical compression by a factor of 2 (or vertical stretch by a
factor of 12 ), down 1 unit.
x
b.
a.
c.
Vertical flip, then up 4 units.
3.
a.
f (x  3)  2
b.

4.
a.
 f (x  1)  1
b.
2 f (x  2)
1
2
f (x)
Section 1.2 Answers
1.
y
2.
a.
–2
c.
y
a.
y  3  56 (x  1.6)
b.
y  1.2  0.6(x  8.1) or y  19.8  0.6(x  7.4)
c.
y  7   54 (x  8)
3.
3
2
x8
1
2
x
5
2
b.
y  2x  200
d.
y
4.
AC  74 , BC  72 ; Point B is closer.
5.
y  23 
7
5
1
2
x2
x  12 
40  2 5  6.32
6.
Lesson 1.3.1 Answers
1.
O  67º , GO  12.88 ft, EO  6.91 ft, area  40.95 ft 2
2.
L  70º , AL  6.80 cm, LG  6.23 cm, area  19.9 cm2
3.
171.56 ft
4.
x  20.74 cm
5.
a.
leg = 5 2 , hyp = 10
b. leg = 2 3 , hyp = 4 3
Lesson 1.3.2 Answers
1.
A  38.2º , B  21.8º , C  120º
2.
NT  32.6, N  82.9º , T  36.1º
3.
51.96 cm 2
4.
52.39 m 2
5.
1 hour = 433.24 miles apart, 2 hours = 866.47 miles apart.
Lesson 1.4.1 Answers
1.
a.
5
12
2.
a.
540
3.
a.
4.
5.
43
3
a.

radians
b.
105º
 171.9º
b.
2
b.
25
18
b.
rad
2.94(2 )  18.46 sec
–108º
6
 1.6 radians
radians
 45 radians
second
1200
130
rev
 2.94 sec
Lesson 1.4.2 Answers
1.
a.
5
3
b.

4
2.
a.

2
b.
2
3
3.
Answers vary. The closest angles are  54 and 114 .
4.
Answers vary. The closest angles are  72 and 2 .
5.
Answers vary. The closest angles are  76 and 56 .
Lesson 2.1.1 Piecewise Functions
for x  1
8
f (x)  
 5x  9 for x  1
1.
Given f(x), evaluate f(–1), f(1), and f(6).
2.
Given h(x), evaluate h(–2), h(0), and h(2). h(x)   1
3.
 x3  3

f (x)   5

 x 1
4.
5.
Graph.
Graph.
 x 2
g(x)  
 x 2
Graph .
x
 x
h(x)   5

2
 (x  1)
1  x 2
for x  0
 2 x  1
for x  0
for x  1
for x  1
for x  1
for x  1
for x  1
for x  1, x  0
for x  0
for x  1
Lesson 2.1.2 Shifting Piecewise Functions and Periodic Functions
 x 4
1.
Let f (x)  
2.
Let h(x)  
2

x  2
3.
Let k(x)  
1
 2 x  1
4.
x  5

 2x  1
 x  2
for x  1
. Graph g(x)  g(x  1)  3 and write an equation for g(x).
for x  1
for x  –1
for x  –1
for x  2
for x  2
. Graph j(x)  j(x  1)  2 and write an equation for j(x).
.
a.
Write an equation, m(x), that will shift k(x) left 2 units and down 5 units.
b.
Is the graph of k(x) continuous? Explain.
Consider the graph of f(x) at right. Assume it continues the pattern
in both the positive and negative directions.
a.
What is the period of the function?
b.
Find 5 x-values where f(x) = 3.
4
f(x)
2
x
c.
5.
2
What is f(23)? Explain how you know.
4
8
6
Consider the graph of g(x) at right. Assume it continues the pattern
in both the positive and negative directions.
4
a.
g(x)
What is the period of the function?
2
b.
Find 2 positive and 2 negative x-values that are not shown on the
graph, where g(x) = –1.
x
2
c.
What is g(41)? How do you know?
4
6
8
Section 2.2 Summation Notation
5
1.
Expand.
 4i 3  1
i2
4
2.
Expand.
 3 j2  5
i1


3.
1  1  1  1
Write the given expression using sigma notation. 0.4 12  2.4
2.8 3.2 3.6
4.
Write the given expression using sigma notation. 0.2 4 3  4 3.2  4 3.4  L  4 4.8
5.
Write the given expression using sigma notation. 2 32.7  32.8  L  33.3  33.4




Section 2.3 Area Under a Curve
1.
Sketch the curve, showing the rectangles used to find the area. Use sigma notation to find the specified area.
Determine if the result is a lower or upper bound for the actual area.
Given f (x)  x 2  3 , find A( f (x), 3  x  6) using 5 left-endpoint rectangles.
2.
Sketch the curve, showing the rectangles used to find the area. Use sigma notation to find the specified area.
Determine if the result is a lower or upper bound for the actual area.
Given g(x) 
x  2 , find A(g(x), 5  x  7) using 10 right-endpoint rectangles.
3.
Sketch the curve, showing the rectangles used to find the area. Use sigma notation to find the specified area.
Determine if the result is a lower or upper bound for the actual area.
Given h(x)  1x , find A(h(x), 1  x  9) using 20 left-endpoint rectangles.
4.
Sketch the curve, showing the rectangles used to find the area. Use sigma notation to find the specified area.
Determine if the result is a lower or upper bound for the actual area.
Given j(x)  13 (x  1)2  4 , find A( j(x),1  x  4) using 5 right-endpoint rectangles.
5.
Given f (x)  x 2  1 , estimate the area under the curve for 2  x  4 using 12 midpoint rectangles.
6.
Given f (x)  x 4  4 , estimate the area under the curve for 0  x  2 using 5 midpoint rectangles.
7.
Given g(x)  2 x , estimate the area under the curve for 0  x  3 using 5 trapezoids.
8.
Given h(x)  2x 2  1 , estimate the area under the curve for 1  x  3 using 5 trapezoids.
9.
If A( f (x), 1  x  3)  17 and h(x)  f (x  2)  5 , what does A(h(x), 3  x  1) equal?
10.
If A( f (x), 2  x  5)  20 and h(x)  f (x  3)  2 , what does A(h(x), 5  x  2) equal?
11.
If A( f (x), 5  x  8)  57 and h(x)  f (x  1)  4 , what does A(h(x), 4  x  7) equal?
12. The area under the curve of the function f (x) from x  1 to x  4 is 24 units2. Find the corresponding area
under the curve for g(x)  f (x  5)  2 . What is the corresponding domain for g(x) ?
13.
Given f (x)  x 3  2 , find A( f (x), 1  x  2) using 6 partitions and the method of your choice. Show all of
your work.
14.
Given j(x) 
your work.
15.
A car travels at 15t  30 mph for 0  t  3 hours. How far does it travel?
16.
A car travels at 12 t 2  55 mph for 0  t  5 hours. Approximately how far does it travel?
x  1  1 , find A( j(x), 1  x  2) using 6 partitions and the method of your choice. Show all of
Lesson 2.1.1 Answers
1.
f (1)  8, f (1)  4, f (6)  21
2.
h(2)  3, h(0)  1, h(2)  2
3.
4.
y
y
f(x)
x
x
5.
y
x
Lesson 2.1.2 Answers
1.
See graph at right.
 (x  1)4  3
g(x)  
 x 1

2.
y
See graph at right.
 2x  1
j(x)  
 (x  1)2

x
for x  0
for x  0
y
for x  0
for x  –1
x
3.
 x  5
m(x)   1
 2 x  5
for x  0
for x  0
Yes, the graph is continuous because both equations in m(x) contain the point (0, –5).
Or – Both equations in k(x) contain the point (2, 0). Shifting a graph does not affect continuity.
4.
a.
3
b. … , –5, –2, 1, 4, 7, …
c.
Because the period is 3, we can subtract any multiple of 3 from a given x-value. 23  3(7)  2 . Therefore,
f (23)  f (2)  1 .
5.
a.
4
c.
Because the period is 4, we can subtract any multiple of 4 from a given x-value. 41  4(10)  1 . Therefore,
g(41)  g(1)  2.5 .
Section 2.2 Answers
1.
31, 107, 255, 499
2.
8, 17, 32, 53
4
3.
1
 Other answers are possible.
 0.4 0.4i2
i0
9
4.
 0.2 4 0.2 j3  Other answers are possible.
j0
7
5.
 2 30.1k 2.7  Other answers are possible.
k 0
b. …, –4, –2 and 10, 12, …
Section 2.3 Answers
  0.6  0.6x  32  3  64.08 u2 , lower bound
4
1.
x0
10
2.
  0.2 
0.2x  5  2   4.04 u 2 , upper bound
x1
19
3.
  0.4 0.41x1   2.39 u2 , upper bound
x0
5
4.
  0.6
x1
11
5.

6.


5
 4    0.6 0.12x 2  4  26.6 u 2 , upper bound
 0.5  0.5x  1.75 2  1 
x0
4

2
1
0.6x  1  1
3
x1
  0.5  0.5x  2.25 2  1  29.88 u2
12
x1
 0.4   0.4x  0.2 2  4  
x0
7.
10.25 u2
8.
15.44 u2
9.
37
10.
34
11.
45
  0.4   0.4x  0.2 2  4   1.81 u2
5
x1
12.
18 u2, 6  x  9
13.
L = –4.3, R = 0.19, M = –2.06, T = –2.06
14.
L = –0.036, R = 0.83, M = 0.4, T = 0.76
15.
157.5 miles
16.
The exact answer (using calculus) is 295.8 miles. Student answers will vary depending on the method they use to
solve the problem.
Lesson 3.1.1 Horizontal and Vertical Stretches
1.
Given f (x) at right, sketch the following transformations.
y
f(x)
a.
2 f (x)
b.
f
12 x 
x
y
2.
Given g(x) at right, sketch the following transformations.
g(x)
x
a.
1
2
g(x)
b.
f (3x)
3.
Given h(x) at right, sketch the following transformations.
y
a.
b.
3h(x)
h(x)
h(2x)
x
4.
Given j(x) at right, write an expression in terms of j(x) the
following transformations.
y
j(x)
x
a.
b.
y
y
x
x
5. Given k(x) at right, write an expression in terms of k(x) the
following transformations.
y
k(x)
x
a.
b.
y
x
y
x
Lesson 3.1.2 Applications of Exponential Functions
1.
Find the equation of the exponential function that passes through the points (2, 36) and (4, 81).
2.
Find the equation of the exponential function that passes through the points (1, 2) and (3, 18).
3.
Find the equation of the exponential function that passes through the points (1, 36) and (3, 64).
4.
A culture begins with 3,800 bacteria. After one hour the count is 10,000. Find a function that models the number
of bacteria after t hours. How many bacteria are present after 6 hours?
5.
The half-life of Uranium 235 is 4.49x108 years. If a disposed fuel rod contain 5 kg of Uranium 235 today, how
much of this radioactive isotope will it contain one million years from now?
6.
The population of a certain country was 2,345,235 in 1985 and 3,458,377 in 1995. Assuming the growth is
exponential, find a formula to model the population t years after 1985. Use the formula to find the time required
for the population to double.
Lesson 3.1.3 Stretching Exponential Functions
1.
Rewrite the given equation in y  a  b x form.
y  3(4)2x1
2.
Rewrite the given equation in y  a  b x form.
y  4(9) 2
3.
Rewrite the given equation in y  a  b x form.
y  4(3)2x2
4.
Use the rules of exponents to express each side with the same base then solve.
a.
5.
322x1 
161 
2x
32  813x 
b.
1 x2
271 
4x1
c.
16 x 
321 
2 x
8
Simplify the following complex fractions.
a.
1  y2
x2
x2  1
y2
b.
2 y
x3
x 1
y2
c.
x 1
y2 x
y
 y2
x2
Lesson 3.2.1 Inverse Functions
1.
Find the inverse of the given function. Is the inverse a function?
f (x) 
2.
Find the inverse of the given function. Is the inverse a function?
f (x)  (x  3)2  4
3.
Find the inverse of the given function. Is the inverse a function?
f (x)  3(4x  1)1/3
4.
Find the inverse of the given function. Is the inverse a function?
f (x)  (2x  1)5/2  3
5x
2x3
Lesson 3.2.2 Logs as Inverse Exponentials
1.
Rewrite each log equation as an exponential equation.
a.
2.
log B M  K
c.
C  logV I
5 2 
1
25
b.
Ay  B
c.
7x  M
log 3 3
b.
log6
b.
2 log2 16
361 
c.
log12 1
c.
4 log 3 27 x
Simplify each log expression.
a.
5.
b.
Evaluate each log expression.
a.
4.
18  3
Rewrite each exponential equation as a log equation.
a.
3.
log2
log 5 (log 2 32)
Solve each equation by rewriting each log equation as an exponential equation.
a.
log 3 9 5   x
b.
log 4 x 
1
2
c.
log x 16  4
Lesson 3.2.3 Graphing Log Functions
1.
Graph f (x)  log 2 (x  3) and determine the domain and range.
2.
Graph f (x)  log 2 x  1 and determine the domain and range.
3.
Graph f (x)   log 3 (x  2) and determine the domain and range.
4.
Graph f (x)  3 log 2 x  1 and determine the domain and range.
Lesson 3.3.1 Laws of Logarithms
1.
Rewrite each of the following expressions using a single logarithm.
a.
2.
1
2
b.
log M  log N  log P 
Use the Laws of Logarithms to expand each of the following log expressions.
a.
3.
2 log M  3 log N
log a
 
x2
yz 7
b.

log m ab 2 c 3

c.

logt h h2  g
Given logb K  1.6, logb J  0.4, and logb L  2.4 find the exact value of:
a.
logb
KL
J3
b.
logb
KL2
Jb 3
c.
 3  JK 2 
logb 

Lb 4 

4.
5.
Simplify.
a.
log 7 (x 2  2x  48)  log 7 (x  8)  log 7 3
b.
2 log 3 (x  2)  log 3 (x  2)  log 3 (x 2  4)
c.
3
2
log2 (x  y)  log(x  y)  log2 (x 2  y2 )
Solve.
a.
log(x  5)  log x  log 5
b.
log 7 (x  4)  log 7 (x  2)  1
c.
log 2 4  log 2 x  log 2 5  log 2 (x  2)
d.
log 3 x 2  log5 25  log2 16
b.
5(2)x  3  47
Lesson 3.3.2 Solving Exponential Equations
1.
Solve.
a.
2.
How could you use the « button on your calculator to evaluate the following log?
a.
3.
log 4 21
b.
log 5 32
3x2  52x
b.
4 x1  72x3
3x
b.
2 x
Solve.
a.
4.
200(1.5)x2  700
Solve.
a.
1.2
8
0.7
59
Lesson 3.1.1 Answers
1.
y
a.
y
b.
x
2.
a.
x
b.
y
y
x
x
y
3.
y
a.
b.
x
x
4.
a.
j
12 x 
b.
3 j(x)
5.
a.
 12 k(x)
b.
k(3x)
Lesson 3.1.2 Answers
1.
y  16
23 
2.
x
y  23  3
3.
y  27
43 
4.
B(t)  3800
5.
4.99 grams
6.
P(t)  2345235(1.04)x
x
x
1950 
x
Lesson 3.1.3 Answers
1.
x
y  43 16 
2.
y
3.
y  36  9 
4.
a.
x
5.
a.
y2  x 2 y 4
x 4 y2  x 2
4  x
3
81
x
5
2
b.
5
x   24
c.
x7
b.
2 y2  y 3 x 3
x 5 y x 3
c.
x 3  xy2
y3  x2 y4
Lesson 3.2.1 Answers
3x
52x
1.
f 1(x) 
2.
f 1 (x)  3  x  4 ; No, unless the range is restricted.
3.
f 1 (x) 
1
4
4.
f 1(x) 
(x3)2/5 1
;
2

3x
2x5
; yes
   1; yes
x 3
3
yes
Lesson 3.2.2 Answers
1.
a.
23 
2.
a.
log5
3.
a.
4.
5.
b.
BK  M
c.
VC  I
b.
log A B  y
c.
log 7 M  x
1
b.
–2
c.
0
a.
1
b.
16
c.
12x
a.
x  10
b.
x2
c.
x
1
8
251  2
1
2
Lesson 3.2.3 Answers
y
1.
D: x > –3
x
3.
D: x > 0
R: all reals
x
D: x >–2
y
y
2.
4.
R: all reals
D: x > 0
y
R: all reals
R: all reals
x
x
Lesson 3.3.1
 
M2
N3
1.
a.
log
2.
a.
2 log a x  log a y  7 log a z
c.
logt h  12 logt (h2  g)
3.
a.
3.2
4.
a. log 7 3(x  6) 
5.
a.
x
c.
x  10
5
4
 
MN
P
b.
log
b.
log m a  2 log m b  23 log m c
b.
3.8
c.
2.4
b.
log 3 (x  2)
c.
1
2
b.
x  1 10
d.
x  27
log(x  y)
Lesson 3.3.2 Answers
1.
a. x  1.09
b.
x  3.32
2.
a.
log 21
log 4
b.
log 32
log 5
3.
a.
x  1.036
b.
x  1.777
4.
a.
x  2.265
b.
x  2.692
Lesson 4.1.1 Special Angles in the Unit Circle
1.
Find the coordinates of the given angle in the unit circle.
a.
2.

3
b.
3
4
c.
5
6
c.
 6
c.
13
4
Find the coordinates of the given angle in the unit circle.
a.
3.

4
3
b.

3
2
Find the coordinates of the given angle in the unit circle.
a.
19 
2
b.
10 
3
Lesson 4.1.2 Sine and Cosine in the Unit Circle
1.
State the reference angle that corresponds to the given angle.
a.
2.
c.
15 
4
cos
6 
b.
sin
54 
c.
cos
c.
sin  34
c.
cos
43 
 
sin  3
b.
 
cos  56
 
Find the exact value of the given trig expression.
a.
5.
19 
6
Find the exact value of the given trig expression.
a.
4.
b.
Find the exact value of the given trig expression.
a.
3.
13
3
cos
113 
b.
sin
176 
174 
Show that the given point is on the unit circle. Then find sin P, cos P, and tan P .
a.
P
1213 ,  135 
b.

P  53 , 45

c.

P 
15
8
,
7
8

Lessons 4.1.3 and 4.1.4 Graphs of Sine and Cosine
1.
Graph the given function.
a.
2.
y  2  sin x
b.


y  2 cos x  4  1
c.
Write an equation for the transformation of y  cos x that has the following properties:
 An amplitude of 5.
 Shifted right 3 units.
 Shifted down 2 units.
3.
Write an equation for the transformation of y  sin x that has the following properties:
 An amplitude of 23 .
 Reflected vertically.
 Shifted up 7 units.
4.
Write an equation involving cosine for the graph shown below.
y
4
2
x
–2
–4
5.
Write an equation involving sine for the graph shown below.
y
4
2
x
–2
–4

y  3sin x  2

Lesson 4.2.1 Reciprocal Trig Functions
1.
Find the exact value of each trig expression.
a.
sec
34 
b.
csc  114

d.
cot
56 
e.
sec(765º )

 
c.
cot  83
f.
csc
233 
2.
If tan   43 and     32 , find the values of the other 5 trig functions. Draw a diagram to assist with the
problem.
3.
If csc    57 and 32    2 , find the values of the other 5 trig functions. Draw a diagram to assist with the
problem.
4.
If sec   83 and 0    2 , find the values of the other 5 trig functions. Draw a diagram to assist with the
problem.
5.
Find all values of  from 0, 2  where the given equation is true.
a.
sin   1
b.
cos  
d.
cos   1
e.
sin   
2
2
2
2
c.
sin   0
f.
cos  
3
2
Lesson 4.2.2 Simplifying Trig Expressions
1.
2.
Simplify the following trig expressions.
a.
sec xcos x
tan x
b.
cos2 x sec2 x  1
c.
sin x csc x
tan x
d.
1sec x
sin xtan x
e.
sin2 x  sec x  cos x
f.
1cos x
1sec x
sec   2
c.
csc  is undefined
4
c.
5
c.
y  2 cos
Find all values of  , where 0    2 , that make the equation true.
a.
3.
cot  
3
3
b.
Simplify the following radical expressions.
a.
3 16x 4 y 5
b.
81m 7 n16
96x16 y 27
Lesson 4.2.3 Frequency of Sine and Cosine Graphs
1.
Find the period of the given trigonometric function.
a.
y  5  3 cos(2 x   )
b.
y  7 sin
23 x  13  4
2 x  1
2.
A bicycle wheel is making 2 revolutions every 3 seconds. What is the angular frequency of the wheel?
3.
What is the angular frequency of the second hand of a clock?
4.
A carousel makes 12 revolutions during a 5-minute ride. What is the angular frequency of the carousel?
5.
Graph the given trig function. State or label all key features of the graph.
a.
y  2 cos(4x)  1
b.

y   sin 2x  2

c.
y  3 cos
2x   
Lesson 4.2.4 Verifying Trig Identities
1.
Verify the given trig identity.
cos x  2 sec x
 1sin
x
a.
cos x  sin x tan x  sec x
b.
1sin x
cos x
c.
 tan x  cot x 2  sec 2 x  csc 2 x
d.
cot x  tan x  sec x csc x
e.
sin x
sec2 x1
f.
 tan x  12  12 sin2x cos x
2
x
 1sin
sin x
cos x
Lesson 4.3.1 Applications of Trig Functions
1.
Write the equation of a sinusoidal function that will model the height of an object that oscillates between 4 feet
above the ground when t  5 seconds and 1 foot above the ground when t  7 seconds.
2.
A weight hanging from a spring oscillates up and down infinitely with the same amplitude. When t  0 the weight
is at its minimum height of 24 inches off of the ground. It takes 2 seconds for the weight to reach its maximum
height of 36 inches off of the ground. Write the equation of a sinusoidal function that models the oscillation of the
weight.
3.
Write the equation of a sinusoidal function that will model the height of a rider on a Ferris wheel. Assume the rider
gets on the Ferris wheel at ground level and it takes 3 minutes to get the top of the wheel at a height of 80 feet.
4.
A piece of gum gets stuck to a bicycle wheel. If the wheel is turning at 2 revolutions per second and has a diameter
of 26 inches, write the equation of a sinusoidal function that will model the situation.
5.
Write the equation of a sinusoidal function that will model the height of the second hand above a horizontal line
thorough the center of a clock. Use time in seconds and assume t  0 is when the second hand in on the 12. The
length of the second hand is 3.5 inches.
Lesson 4.1.1 Answers
1.
a.
, 
2.
a.

3.
a.
0, 1
3
2
1
2
1
2
,
3
2

b.

c.

3
2
b.
0,1
c.

3
2
,
b.

c.

2
2
,
b.

6
c.

4
b.

2
2
c.

b.

3
2
c.

b.
1
2
2
2
1
2
,
,
2
2
3
2


Lesson 4.1.2 Answers

3
1.
a.
2.
a.
3.
a.

4.
a.
1
2
5.
a.
1
1213    135   14425
169
3
2
3
2
2
2
5 , cos P  12 , tan P   5
sin P   13
13
12
c.
1
2
2
2
2
2
, 12
1
2
2
2



b.
1
 53   45   916
25
2
sin P 
c.
2
4
5
, cos P   53 , tan P  
     
15
8
2
7 2
8
1549
64
sin P   87 , cos P  
15
8
4
3
1
, tan P 
7 15
15
Lesson 4.1.3 and 4.1.4 Answers
1.
a.
y
2
1
x
–1
–2
b.
y
4
2
x
–2
–4
c.
y
4
2
x
–2
–4
2.
y  5 cos(x  3)  2
3.
y   23 sin(x)  7
4.
y  3 cos(x   )  1 or y  3 cos(x)  1
5.
y  4 sin x  2  2 or y  4 sin x  2  2




Lesson 4.2.1 Answers
1.
a.
 2
b.
 2
c.
d.
 3
e.
2
f.
 2 33
c.
0, 
f.

2.
sin    45 , cos    53 , csc    54 , sec    53 , cot  
3.
sin    57 , cos  
4.
sin  
5.
a.
d.
3
2
0
55
8
2 6
7
, tan    5126 , sec  
, cos   83 , tan  
55
3
, csc  
b.

e.
5
4
4
,
8 55
55
7
4
,
7
4
7 6
12
3
3
3
4
, cot    2 5 6
, cot  
3 55
55
6
,
11
6
Lesson 4.2.2 Answers
a.
sin x
b.
sin 2 x
c.
cot x
d.
csc x
e.
sec x
f.
cos x
2.
a.

b.
2
3
c.
0, 
3.
a.
2xy 3 3 2xy 2
b.
3mn 4 4 m 3
c.
2x 3 y 5 5 3xy 2
b.
3
c.
4
3.

30
4.
24 
5
1.
3
,
4
3
,
4
3
Lesson 4.2.3 Answers
1.
2.
5.
a.
1
4
3
a.
y
4
2
x

2
Ğ2
3
Ğ4
b.
y
4
2
x

Ğ2
Ğ4
2

3
2
2
c.
y
4
2
x
4
2
Ğ2
6
Ğ4
Lesson 4.2.4 Answers
1.
a.
cos x  sin x tan x  sec x
cos x  sin x
sin x
cos x
2x
cos x  sin
cos x
cos2 xsin 2 x
cos x
1
cos x
b.



 sec x
1sin x
cos x
cos x  2 sec x
 1sin
x
(1sin x)2 cos2 x
cos x(1sin x)

12 sin xsin 2 xcos2 x
cos x(1sin x)

12 sin x1
cos x(1sin x)

2(1sin x)
cos x(1sin x)

2
cos x
c.
 tan x  cot x 2  sec 2 x  csc 2 x
tan 2 x  2 tan x cot x  cot 2 x 
tan 2 x  2  cot 2 x 
sin 2 x
cos2 x
cos2 x
sin 2 x

sin 4 x2 sin 2 x cos2 xcos 4 x
cos2 x sin2 x 
(sin 2 xcos2 x)2

2
cos2 x sin2 x 

1  1
cos2 x
sin 2 x
sin 2 xcos2 x
cos2 x sin2 x 
1
cos2 x sin2 x 

1
cos2 x sin2 x 
 2 sec x
d.
cot x  tan x  sec x csc x
sin x
x
 cos
cos x
sin x
sin 2 xcos2 x
 cos x  sin x 
1
 cos x  sin x 
e.


sin
 sec x csc x
x
 1sin
sin x
cos2 x
sin x
x
 1sin
sin x
 tan x  12  12 sin 2x cos x
f.
cos x
tan 2
x  2 tan x  1 
sec 2 x  2 tan x 
1
cos2 x

2 sin x
cos x
 12 sin 2x cos x
cos x
Lesson 4.3.1 Answers
2  t  4  2.5
f (t)  1.5 cos 2  t  5   2.5
1.
f (t)  1.5 sin
2.
f (t)  6 sin
3.
f (t)  40 sin
4.
f (t)  13 sin 4 t 
5.
2  t  1 30
f (t)  6 cos 2  t  2   30
2  t  3 30
f (t)  6 cos 2 t  30
   13
f (t)  13 cos 4 t   13
3  t  4.5  40
f (t)  40 cos 3 t  40
f (t)  40 sin
   13
1
8
f (t)  13 sin 4 t 
1
4
f (t)  13 cos  4 t   13
30  t  45 
 t
f (t)  3.5 cos 30

f (t)  3.5 sin
2  t  2  2.5
f (t)  1.5 cos 2  t  3   2.5
f (t)  1.5 sin
f (t)  6 sin
3  t  1.5  40
f (t)  40 cos 3  t  3   40
3
8
3  t  15 
f (t)  3.5 cos 3  t  30  
f (t)  3.5 sin
2
sin x
sec2 x1
sin x
tan 2 x
2
x  cos2 x
sin x


2
Lesson 5.1.1 Inverse and Direct Variation
1.
If y varies directly as (x  1) and f (4)  4 , find f (2) .
2.
If y varies directly as  x 2  2x  and f (5)  9 , find f (1) .
3.
If y varies inversely as (x  1) and f (3)  59 , find f (3) .
4.
If y varies inversely as  x 3  1 and f (1)  47 , find f 12 .

Lesson 5.1.2 Transformations of Rational Functions
1.
1 1
Sketch the graph of f (x)  x2
.
2.
1 2
Sketch the graph of f (x)  x4
.
3.
Rewrite f (x)  2x5
as a transformation of g(x)  1x and sketch the graph of f (x) .
x2
4.
Rewrite f (x)  3x7
as a transformation of g(x)  1x and sketch the graph of f (x) .
x3
5.
Write the equation of a rational function that has a horizontal asymptote at y  13 and a vertical asymptote at
x  2 .
6. Write the equation of a rational function that has a horizontal asymptote at y  12 and a vertical asymptote at x  9
Lesson 5.1.3 Graphing Reciprocals of Functions
1.
1 .
Given f (x) shown at right, sketch the graph of f (x)
y
f(x)
y
x
x
2.
1 .
Given f (x) shown at right, sketch the graph of f (x)
y
f(x)
y
x
x
3.
1 .
Given f (x) shown at right, sketch the graph of f (x)
y
f(x)
y
x
x
4.
1 has vertical asymptotes at x  4 and x  3 .
Give an example of a function f (x) such that f (x)
5.
1 has vertical asymptotes at x  6 and x  5 .
Give an example of a function f (x) such that f (x)
Lessons 5.2.1 and 5.2.2 Introduction to Limits
1.
Evaluate the following limits. If the limit does not exist, explain why.
a.
d.
2.
 
b.

e.
lim 5
x x4
lim 2
x x
1

c.
lim 3x 2 
f.
 x3 
lim 
x
x
lim f (x)
b.
x
lim  cos x 
x
lim f (x)
x2
c.
lim f (x)
x2
5 1
Sketch the graph of f (x)  x3
. Use it to evaluate the limit statements below.
a.
lim f (x)
b.
x
4.
 
1 2
Sketch the graph of f (x)  x2
. Use it to evaluate the limit statements below.
a.
3.
lim 2x1
x 2x1
lim f (x)
x3
c.
lim f (x)
x3
5x
a k
Given f (x)  x2
, rewrite the function in the form y  xh
. Then evaluate the following limit statements.
a.
lim f (x)
x
b.
lim f (x)
x2 
c.
lim f (x)
x2 
5.
a  k . Then evaluate the following limit statements.
Given f (x)  3x1
, rewrite the function in the form y  xh
x1
a.
b.
lim f (x)
x
c.
lim f (x)
x1
lim f (x)
x1
Lesson 5.2.3 Working With One-Sided Limits
1.
x
x
Let f (x)  4 3
.
x
Complete the table of values below to estimate the value of lim
4 x  3x
x
x0
x
–0.01
–0.001
–0.0001
0.0001
.
0.001
0.01
f (x)
2.
2
Let f (x)  x 3x10
.
x2
Complete the table of values below to estimate the value of lim
x 2  3x10
x2
x2
x
1.9
1.99
1.999
2.001
.
2.01
2.1
2.01
2.1
f (x)
3.
3
8
Let f (x)  xx2
.
Complete the table of values below to estimate the value of lim
x 3 8
x2 x2
x
f (x)
1.9
1.99
1.999
2.001
.
4.
Use a graph or a table to evaluate the following limits.
a.
5.
1
x x2
lim
5
b.
x 2 6
x4 x1
c.
lim
lim
x5 
2
x5
1
Use a graph or a table to evaluate the following limits.
a.
1
2
x1
x
lim
3
b.
x 2 1
x2 x1
3
x1

x1
c.
lim
lim
2
Lessons 5.2.4 and 5.2.5 Continuity, Piecewise Functions, and Limits
1.
2.
Determine whether the given function is continuous at the point specified and then determine is the limit exists.
Explain your answers.
a.
f (x)  3x2  1, x  2
b.
f (x) 
2x1 
x4
c.
 x2  1
for x  3
f (x)  
, x3
 2(x  1) for x  3

d.
f (x) 
x 2 25
x5
3, x  4
 1, x  5
Find values for m and n such that f (x) will be a continuous function.
a.
 3x  m

f (x)   2x 2  4
 3x  n
for x  1
for  1  x  1
for x  1
b.
 12 x  m

f (x)   x 2  8
 2x  n

for x  4
for 4  x  7
for x  7
3.
Given the piecewise function f (x) shown below, evaluate the following expressions.
a.
f (6)
y
2
b.
c.
x
lim f (x)
x6
2
lim f (x)
x3
d.
e.
f (7)
lim f (x)
f.
x 7
4.
g.
lim f (x)
lim f (x)
x
x
Given the piecewise function f (x) shown below, evaluate the following expressions.
y
a.
f (6)
2
x
b.
2
lim f (x)
x6
c.
d.
e.
lim f (x)
x4 
lim f (x)
x4 
f (4)
f.
lim f (x)
x
g.
lim f (x)
x
5.
2x  1
Let f (x)  
 2x  1
a.
for x  3 .
for x  3
Find lim f (x)
b.
x3
6.
 x 3  2x  1
Let f (x)   4 x7

a.
x1
Find lim f (x)
x1
lim f (x)
x0
for x  1
.
for x  1
b.
lim f (x)
x2
Lesson 5.1.1 Answers
1.
–4
3.

5
8
2.

4.
4
9
35
Lesson 5.1.2 Answers
1.
2.
y
y
x
3.
f (x) 
1
x2
2
x
4.
f (x) 
2
x3
3
y
y
x
x
5.
f (x) 
a
x2
 13
6.
f (x) 
a
x9
 12
Lesson 5.1.3 Answers
1.
2.
y
y
x
3.
x
See graph at right.
y
4.
f (x)  a(x  4)(x  3)
5.
f (x)  a(x  6)(x  5)
x
Lessons 5.2.1 and 5.2.2 Answers
a.
0
b.
2
c.
DNE / 
d.
1
e.
DNE / 
f.
DNE
2.
a.
–2
b.

c.

3.
a.
1
b.

c.

4.
a.
5
b.

c.

5.
a.
3
b.

c.

1.
Lesson 5.2.3 Answers
1.
2.
3.
0.28768
x
–0.01
–0.001
–0.0001
0.0001
0.001
0.01
f (x)
0.28413
0.28732
0.28765
0.28772
0.28804
0.29128
x
1.9
1.99
1.999
2.001
2.01
2.1
f (x)
6.9
6.99
6.990
7.001
7.01
7.1
x
1.9
1.99
1.999
2.001
2.01
2.1
f (x)
11.41
11.94
11.994
12.006
12.06
12.61
7
12
4.
a.
5
b.
2
c.

5.
a.
3
b.
–5
c.

Lesson 5.2.4 and 5.2.5 Answers
1.
a.
Continuous, limit exists.
b.
Not continuous, limit DNE.
c.
Continuous, limit exists.
d.
Not continuous, limit = –10.
2.
a.
m  1, n  1
b.
m  6, n  27
3.
a.
–2
b.

c.
1
e.
–3
f.

g.
–2
d.
1
4.
a.
–2
b.
3
c.
1
e.
–1
f.
1
g.

5.
a.
7
b.
0
6.
a.
DNE
b.
 13
d.
–1