lesson 13.1.1 circles - chagrin falls exempted village … sem 1 review3.pdf · lesson 13.1.2...

PRE-CALCULUS

Semester 1 Review

Chapters 13 and 1 thru 5

Lesson 13.1.1 Circles

1. 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

2. 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. x2 y2 16x 12y 88 d. x2 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. (x3)2

16

(y2)2

25 1 b.

(x4)2

8

(x2)2

4 1

c. 9x2 16y2 144 d. 12x2 9y2 108

e. 4x2 9y2 16x 20 0 f. x2 6y2 4x 60y 136 0

g. 12x2 5y2 24x 30y 39 0 h. 136

x2 125

y2 12

x 925

y 15350

1

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.

a. (x2)2

9

(y2)2

25 1 b.

(y1)2

36

(x3)2

16 1

c. 4x2 8x 12 4y2 d. 4x2 y2 8x 6y 21 0

e. x2 4y2 2x 16y 51 f. y2 4x2 2y 16x 19

g. 2y2 4x2 12y 8x 6 0 h. x2 16y2 4x 160y 400 0

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. y 116

x2

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 x2 2x 3 h. x2 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. 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. x 72

2 y 3

2 2 9

2

g. (x 3)2 (y 1)2 25 h. (x 8)2 (y 9)2 64

2. 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


1. a. C (3, 2)

M 10, m 8

F (3, 5), (3,1)

b. C (4, 2)

M 4 2, m 4

F (6, 2), (2, 2)

c. C (0, 0)

M 8, m 6

F ( 7, 0), ( 7, 0)

d. C (0, 0)

M 4 3, m 6

F (0, 3), (0, 3)

x

y

x

y

x

y

x

y

x

y

e. C (2, 0)

M 6, m 4

F (2 5, 0)

f. C (2, 5)

M 6 2, m 2 3

F (2 15, 5)

g. C (1, 3)

M 4 3, m 2 5

F (1, 3 7)

h. C (9, 4.5)

M 12, m 10

F (9 11, 4.5)

*Graph is scaled by 3’s.

2. (x3)2

9

(y3)2

25 1

3. (x1)2

36

(y2)2

11 1

4. (x2)2

49

(y5)2

33 1

5. (x5)2

25

(y4)2

16 1


1. a. C (2, 2)

y 53

(x 2) 2

F (8, 2), (4, 2)

b. C (3,1)

y 32

(x 3)1

F (3,1 2 13)

*Graph is scaled by 2’s. *Graph is scaled by 2’s.

c. C (1, 0)

y (x 1)

F (1 2 2, 0)

d. C (1, 3)

y 2(x 1) 3

F (1 2 5, 3)

x

y

x

y

x

y

x

y

x

y

x

y

x

y

e. C (1, 2)

y 12

(x 1) 2

F (1 3 5, 2)

f. C (2,1)

y 2(x 2)1

F (2,1 5)


g. C (1, 3)

y 2(x 1) 3

F (1, 3 6)

h. C (2, 5)

y 14

(x 2) 5

F 2 172

, 5

2. y2

16 x2

4 1

3. (y2)2

4

(x3)2

12 1

4. (y2)2

1

(x1)2

3 1

5. x2

9

(y4)2

7 1


1. a. V (0, 0)

F (5, 0)

D : x 5

b. V (0, 0)

F (0, 4)

D : y 4


x

y

x

y

x

y

x

y

x

y

x

y

c. C (3,1)

F (3, 4)

D : y 2

d. V (5, 3)

F (5, 7)

D : y 1


e. V (5, 4)

F 5, 4 14

D : y 3 34

f. V (2, 4)

F (3, 4)

D : x 1


g. V (1, 2)

F 1, 2 14

D : y 1 34

h. V (2, 4)

F (2, 8)

D : y 0


2. x 14

(y 1)2 6

3. y 128

(x 6)2 2

4. x 120

(y 4)2 2

5. y 2(x 4)2 3

x

y

x

y

x

y

x

y

x

y

x

y

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 Weight Height Weight

63 130 71 150

69 155 65 130

73 195 70 160

64 150 62 120

72 185 67 155

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.

2. 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)

Weight

(grams)

Packaging

(in2)

Weight

(grams)

47 28 125 566

138 850 69 85

100 283 88 198

111 425

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)


(1, 2), (2, 3), (3, 1), (3, 3), (5, 7), (6, 8), (8, 6), (9, 9)


(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) x2 , sketch the original function following transformations on the same set of axes.

a. f (x 2) b. f (x) 3 c. 2 f (x)

2. For the parent graph f (x) x , sketch the original function following transformations on the same set of axes.

a. f (x 1) b. 3 f (x) 2 c. f (x 2) 4

3. For the parent graph f (x) x , sketch the original function following transformations on the same set of axes.

a. 2 f (x 3) b. 12

f (x 1) 5 c. 23

f (x 2) 2

4. Given the following function, state the equation of the parent graph, sketch the graph, and describe the

transformation completely.

f (x) 1x5

3

5. 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) x2 8x and g(x) 3x 1 , find and simplify the following function operations.

a. 2 f (x) g(x) b. f (x) g(x) c. g( f (x))

2. Given f (x) 2x2 1 and g(x) 2x 3 , find and simplify the following function operations.

a. f (x 2) b. f (g(x)) c. g( f (x))

3. If f (x) 3x 4 and g(x) x2 3x 1 , find g( f (x)) and simplify completely.

4. Find the inverse of g(x) 2x 1 .

5. Find the inverse of f (x) 4(2x 3)1/3 and simplify completely.

Lesson 1.1.4 Transformations of Non-Parent Graphs

1. Given the graph of m(x), sketch the following transformations.

a. 2m(x)

b. m(x 2)1

c. m(x 1) 3

x

y

m(x)

2. Given the graph of h(x), sketch the following transformations.

a. h(x 3) 2

b. 12

h(x)1

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).

a. b.

x

y

h(x)

x

y

f(x)

x

y

x

y

4. Given the graph of f(x) at right, write an expression for each transformation

graph below in terms of f(x).

a. b.

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 .

x

y

f(x)

x

y

x

y

3. 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).

4. 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 8cm.

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.

5. Find the length of the missing leg and hypotenuse of the special right triangles below.

a. b.

11 cm

26º x

5 2

45

6

30

Lesson 1.3.2 Law of Cosines

1. Solve the triangle at right.

2. Solve the triangle at right.

3. Find the area of the triangle below. Show all of your steps.

3

5

7

A

B

C

I

N

T

37

22

61º

15 cm

13 cm

8 cm

4. Find the area of the triangle below. Show all of your steps.

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. 75º b. 712

radians


a. 3 radians b. 30 º

10 m

12 m

19 m


a. 35

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. 3

b. 94

2. 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. 156

b. 43

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

.


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 559

x 253 .

b. 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

y 8.66x 491.26 . Another possibility using the points (69, 85) and (125, 566) is y 7 1966

x 507 23

.

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 135

.


y 0.89x 0.75 . Another possibility using the points (2, 3) and (9, 9) is y 67

x 97

.


y 1.07x 1.42 . Another possibility using the points (0, 2) and (5, 7) is y 1x 2 .


1. See graph at right.

a. Right 2 units.

b. Down 3 units.

c. Vertical stretch by a factor of 2.


a. Vertical flip, up one unit.

b. Vertical stretch by a factor of 3, down 2 units.

c. Left 2 units, down 4 units.


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.

c. Vertical stretch by a factor of 23

, right 2 units, up 2 units.

x

y a.

b.

c.

x

y

a.

b.

c.

x

y

a.

b.

c.

4. See graph below. 5. See graph below.

Parent graph: y 1x

Parent graph: y x3

Right 5 units, up 3 units. Vertical stretch by a factor of 3,

vertical flip, right 1 unit, down

4 units.


1. a. 2x2 19x 1 b. 3x3 25x2 8x c. 3x2 24x 1

2. a. 2x2 8x 9 b. 4x 5 c. 4x2 1

3. 9x2 15x 3

4. g1(x) x212

5. f 1(x) x3

128 3

2

x

y

x

y



a. Vertical stretch by a factor of 2, vertical flip.

b. Left 2 units, down 1 unit.

c. Right 1 unit, up three units.


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.

c. Vertical flip, then up 4 units.

3. a. f (x 3) 2 b. 12

f (x)

4. a. f (x 1)1 b. 2 f (x 2)

x

y

a.

b.

c.

x

y

a.

b.

c.

Section 1.2 Answers

1. y 32

x 8

2. a. –2 b. y 2x 200

c. y 12

x 52

d. y 12

x 2

3. 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)

4. AC 74 , BC 72 ; Point B is closer.

5. y 32 7

5x 1

2

6. 40 2 5 6.32


1. O 67º , GO 12.88 ft, EO 6.91ft, area 40.95 ft2

2. L 70º , AL 6.80 cm, LG 6.23cm, 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


1. A 38.2º , B 21.8º , C 120º

2. NT 32.6, N 82.9º , T 36.1º

3. 51.96 cm2

4. 52.39 m2

5. 1 hour = 433.24 miles apart, 2 hours = 866.47 miles apart.


1. a. 512

radians b. 105º

2. a. 540

171.9º b. 2

61.6 radians

3. a. –108º b. 2518

radians

4. 433 45 radians

second

5. a. 1200130

2.94 revsec

b. 2.94(2 ) 18.46 radsec


1. a. 53

b. 4

2. a. 2

b. 23

3. Answers vary. The closest angles are 54

and 114

.


and 2

.


and 56

.

Lesson 2.1.1 Piecewise Functions

1. Given f(x), evaluate f(–1), f(1), and f(6). f (x) 8 for x 1

5x 9 for x 1

2. Given h(x), evaluate h(–2), h(0), and h(2). h(x) 1 x2 for x 0

12

x 1 for x 0

3. Graph. f (x)

x3 3 for x 1

5 for x 1

x 1 for x 1

4. Graph. g(x) x2 for x 1

x2 for x 1

5. Graph . h(x)

x

x for x 1, x 0

5 for x 0

(x 1)2 for x 1

Lesson 2.1.2 Shifting Piecewise Functions and Periodic Functions

1. Let f (x) x4 for x 1

x 5 for x 1

. Graph g(x) g(x 1) 3 and write an equation for g(x).

2. Let h(x) 2x 1 for x –1

x2 2 for x –1

. Graph j(x) j(x 1) 2 and write an equation for j(x).

3. Let k(x) x 2 for x 2

12

x 1 for x 2

.

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.

4. 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.

c. What is f(23)? Explain how you know.

5. Consider the graph of g(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 2 positive and 2 negative x-values that are not shown on the

graph, where g(x) = –1.

c. What is g(41)? How do you know?

2

4

x

f(x)

2 4 6 8

2

4

x

g(x)

2 4 6 8

Section 2.2 Summation Notation

1. Expand. 4i3 1i2

5

2. Expand. 3 j2 5i1

4

3. Write the given expression using sigma notation. 0.4 12 1

2.4 1

2.8 1

3.2 1

3.6

4. Write the given expression using sigma notation. 0.2 43 43.2 43.4 L 44.8

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) x2 3 , find A( f (x), 3 x 6) using 5 left-endpoint rectangles.



Given g(x) x 2 , find A(g(x), 5 x 7) using 10 right-endpoint rectangles.



Given h(x) 1x

, find A(h(x),1 x 9) using 20 left-endpoint rectangles.



Given j(x) 13

(x 1)2 4 , find A( j(x),1 x 4) using 5 right-endpoint rectangles.

5. Given f (x) x2 1 , estimate the area under the curve for 2 x 4 using 12 midpoint rectangles.

6. Given f (x) x4 4 , estimate the area under the curve for 0 x 2 using 5 midpoint rectangles.

7. Given g(x) 2x , estimate the area under the curve for 0 x 3 using 5 trapezoids.

8. Given h(x) 2x2 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?



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) x3 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) x 1 1, find A( j(x),1 x 2) using 6 partitions and the method of your choice. Show all of

your work.

15. A car travels at 15t 30 mph for 0 t 3hours. How far does it travel?

16. A car travels at 12

t2 55 mph for 0 t 5 hours. Approximately how far does it travel?


1. f (1) 8, f (1) 4, f (6) 21

2. h(2) 3, h(0) 1, h(2) 2

3. 4.

x

y

f(x)

x

y

5.



g(x) (x 1)4 3 for x 0

x 1 for x 0


j(x) 2x 1 for x 0

(x 1)2 for x –1

3. m(x) x 5 for x 0

12

x 5 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.

x

y

x

y

x

y

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 b. …, –4, –2 and 10, 12, …

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

3. 0.4 10.4i2

i0

4

Other answers are possible.

4. 0.2 40.2 j3 j0

9


5. 2 30.1k2.7 k0

7


Section 2.3 Answers

1. 0.6 0.6x 3 2 3

x0

4

64.08 u2 , lower bound

2. 0.2 0.2x 5 2 x1

10

4.04 u2 , upper bound

3. 0.4 10.4x1

x0

19


4. 0.6 13

0.6x 11 2 4

x1

5

0.6 0.12x2 4 x1

5


5. 0.5 0.5x 1.75 2 1

x0

11

0.5 0.5x 2.25 2 1

x1

12

29.88 u2

6. 0.4 0.4x 0.2 2 4

x0

4

0.4 0.4x 0.2 2 4

x1

5

1.81u2

7. 10.25 u2

8. 15.44 u2

9. 37

10. 34

11. 45

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.

a. 2 f (x) b. f 12

x

2. Given g(x) at right, sketch the following transformations.

a. 12

g(x) b. f (3x)

x

y

f(x)

x

y

g(x)

3. Given h(x) at right, sketch the following transformations.

a. 3h(x) b. h(2x)

4. Given j(x) at right, write an expression in terms of j(x) the

following transformations.

a. b.

5. Given k(x) at right, write an expression in terms of k(x) the

following transformations.

a. b.

x

y

h(x)

x

y

j(x)

x

y

x

y

x

y

k(x)

x

y

x

y

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).



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 bx form. y 3(4)2x1

2. Rewrite the given equation in y a bx form. y 4(9)12

x2

3. Rewrite the given equation in y a bx form. y 4(3)2x2

4. Use the rules of exponents to express each side with the same base then solve.

a. 322x1 116

2x b. 32 813x 1

27 4x1

c. 16x 132

2x 8

5. Simplify the following complex fractions.

a.

1

x2y2

x2 1

y2

b.

2

x3y

x 1

y2

c.

x

y2 1

x

y

x2y2

Lesson 3.2.1 Inverse Functions

1. Find the inverse of the given function. Is the inverse a function? f (x) 5x2x3

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

Lesson 3.2.2 Logs as Inverse Exponentials

1. Rewrite each log equation as an exponential equation.

a. log218 3 b. logB M K c. C logV I

2. Rewrite each exponential equation as a log equation.

a. 52 125

b. Ay B c. 7x M

3. Evaluate each log expression.

a. log3 3 b. log6136 c. log12 1

4. Simplify each log expression.

a. log5(log2 32) b. 2log2 16 c. 4 log3 27x

5. Solve each equation by rewriting each log equation as an exponential equation.

a. log3 95 x b. log4 x 12

c. logx 16 4

Lesson 3.2.3 Graphing Log Functions

1. Graph f (x) log2(x 3) and determine the domain and range.

2. Graph f (x) log2 x 1 and determine the domain and range.

3. Graph f (x) log3(x 2) and determine the domain and range.

4. Graph f (x) 3log2 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 log M 3log N b. 12

log M log N log P

2. Use the Laws of Logarithms to expand each of the following log expressions.

a. logax2

yz7 b. logm ab2 c3 c. logt h h2 g

3. 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

Jb3 c. logb

JK 23

Lb4

4. Simplify.

a. log7(x2 2x 48) log7(x 8) log7 3

b. 2 log3(x 2) log3(x 2) log3(x2 4)

c. 32

log2(x y) log(x y) log2(x2 y2 )

5. Solve.

a. log(x 5) log x log 5 b. log7(x 4) log7(x 2) 1

c. log2 4 log2 x log2 5 log2(x 2) d. log3 x2 log5 25 log2 16

Lesson 3.3.2 Solving Exponential Equations

1. Solve.

a. 200(1.5)x2 700 b. 5(2)x 3 47

2. How could you use the « button on your calculator to evaluate the following log?

a. log4 21 b. log5 32

3. Solve.

a. 3x2 52x b. 4x1 72x3

4. Solve.

a. 3 x 1.2 8 b. 2 x 0.7

5 9


1. a. b.

2. a. b.

3. a. b.

4. a. j 12

x b. 3 j(x)

5. a. 12

k(x) b. k(3x)

x

y

x

y

x

y

x

y

x

y

x

y


1. y 16 32

x 2. y 2

33 x

3. y 27 43

x 4. B(t) 3800 50

19 x

5. 4.99 grams 6. P(t) 2345235(1.04)x


1. y 34

16 x

2. y 481

3 x

3. y 36 9 x

4. a. x 52

b. x 524

c. x 7

5. a. y2 x2y4

x4 y2 x2 b.

2y2 y3x3

x5yx3 c.

x3xy2

y3x2y4


1. f 1(x) 3x52x

3x2x5

; yes

2. f 1(x) 3 x 4 ; No, unless the range is restricted.

3. f 1(x) 14

x3

31 ; yes

4. f 1(x) (x3)2/51

2; yes


1. a. 23 18

b. BK M c. VC I

2. a. log5125 2 b. logA B y c. log7 M x

3. a. 1 b. –2 c. 0

4. a. 1 b. 16 c. 12x

5. a. x 10 b. x 2 c. x 12


1. D: x > –3 2. D: x > 0

R: all reals R: all reals

3. D: x >–2 4. D: x > 0

R: all reals R: all reals

Lesson 3.3.1

1. a. log M 2

N 3 b. log MNP

2. a. 2 loga x loga y 7 loga z b. logm a 2 logm b 32

logm c

c. logt h 12

logt (h2 g)

3. a. 3.2 b. 3.8 c. 2.4

4. a. log7 3(x 6) b. log3(x 2) c. 12

log(x y)

5. a. x 54

b. x 1 10

c. x 10 d. x 27

x

y

x

y

x

y

x

y


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. 3

b. 34

c. 56


a. 43

b. 32

c. 6


a. 192

b. 103

c. 134

Lesson 4.1.2 Sine and Cosine in the Unit Circle

1. State the reference angle that corresponds to the given angle.

a. 133

b. 196

c. 154

2. Find the exact value of the given trig expression.

a. cos 6 b. sin 5

4 c. cos 43


a. sin 3 b. cos 5

6 c. sin 34


a. cos 113 b. sin 17

6 c. cos 174

5. Show that the given point is on the unit circle. Then find sin P, cos P, and tan P .

a. P 1213

, 513 b. P 3

5, 4

5 c. P 158

, 78

Lessons 4.1.3 and 4.1.4 Graphs of Sine and Cosine

1. Graph the given function.

a. y 2 sin x b. y 2 cos x 4 1 c. y 3sin x

2

2. 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.

5. Write an equation involving sine for the graph shown below.

x

y

2

–2

–4

4

x

y

2

–2

–4

4

Lesson 4.2.1 Reciprocal Trig Functions

1. Find the exact value of each trig expression.

a. sec 34 b. csc 11

4 c. cot 83

d. cot 56 e. sec(765º ) f. csc 23

3

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 75

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 22

c. sin 0

d. cos 1 e. sin 22

f. cos 32

Lesson 4.2.2 Simplifying Trig Expressions

1. Simplify the following trig expressions.

a. sec xcos xtan x

b. cos2 x sec2 x 1 c. sin x csc xtan x

d. 1sec xsin x tan x

e. sin2 x sec x cos x f. 1cos x1sec x

2. Find all values of , where 0 2 , that make the equation true.

a. cot 33

b. sec 2 c. csc is undefined

3. Simplify the following radical expressions.

a. 16x4y53 b. 81m7n164 c. 96x16y275

Lesson 4.2.3 Frequency of Sine and Cosine Graphs

1. Find the period of the given trigonometric function.

a. y 5 3cos(2x ) b. y 7 sin 23

x 13 4 c. y 2 cos

2x 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 3cos x

2

Lesson 4.2.4 Verifying Trig Identities

1. Verify the given trig identity.

a. cos x sin x tan x sec x b. 1sin xcos x

cos x1sin x

2 sec x

c. tan x cot x 2 sec2 x csc2 x d. cot x tan x sec x csc x

e. sin x

sec2 x1 1sin2 x

sin x f. tan x 1 2 12 sin x cos x

cos2 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.


1. a. 12

, 32 b. 2

2, 2

2 c. 32

, 12

2. a. 12

, 32 b. 0,1 c.

32

, 12

3. a. 0, 1 b. 12

, 32 c.

22

, 22


1. a. 3

b. 6

c. 4

2. a. 3

2 b. 2

2 c. 1

2

3. a. 32

b. 32

c. 22

4. a. 12

b. 12

c. 2

2

5. a. 1213

2 5

13 2 14425

169 1

sin P 513

, cos P 1213

, tan P 512

b. 35

2 4

5 2 916

25 1

sin P 45

, cos P 35

, tan P 43

c. 158

2

78

2 1549

64 1

sin P 78

, cos P 158

, tan P 7 1515

Lesson 4.1.3 and 4.1.4 Answers

1. a.

b.

c.

x

y

1

–1

–2

2

x

y

2

–2

–4

4

x

y

2

–2

–4

4

2. y 5 cos(x 3) 2

3. y 23

sin(x) 7

4. y 3cos(x )1 or y 3cos(x)1

5. y 4 sin x 2 2 or y 4 sin x

2 2


1. a. 2 b. 2 c. 3

3

d. 3 e. 2 f. 2 33

2. sin 45

, cos 35

, csc 54

, sec 53

, cot 34

3. sin 57

, cos 2 67

, tan 5 612

, sec 7 612

, cot 2 65

4. sin 558

, cos 38

, tan 553

, csc 8 5555

, cot 3 5555

5. a. 32

b. 4

, 74

c. 0,

d. 0 e. 54

, 74

f. 6

, 116


1. a. sin x b. sin2 x c. cot x

d. csc x e. sec x f. cos x

2. a. 3

, 43

b. 23

, 43

c. 0,

3. a. 2xy3 2xy23 b. 3mn4 m34 c. 2x3y5 3xy25


1. a. 1 b. 3 c. 4

2. 43

3. 30

4. 245

5. a.

b.

x

y

3 2

2

Ğ2

Ğ4

4

x

y

2

32

2

2

Ğ2

Ğ4

4

c.


1. a. cos x sin x tan x sec x

cos x sin x sin xcos x

cos x sin2 xcos x

cos2 xsin2 xcos x

1cos x

sec x

b. 1sin xcos x

cos x1sin x

2 sec x

(1sin x)2 cos2 x

cos x(1sin x)

12 sin xsin2 xcos2 xcos x(1sin x)

12 sin x1cos x(1sin x)

2(1sin x)

cos x(1sin x)

2cos x

2 sec x

c. tan x cot x 2 sec2 x csc2 x

tan2 x 2 tan x cot x cot2 x 1

cos2 x 1

sin2 x

tan2 x 2 cot2 x sin2 xcos2 x

cos2 x sin2 x

sin2 x

cos2 x 2 cos2 x

sin2 x 1

cos2 x sin2 x

sin4 x2 sin2 x cos2 xcos4 x

cos2 x sin2 x

(sin2 xcos2 x)2

cos2 x sin2 x 1

cos2 x sin2 x

x

y

6 2 4

2

Ğ2

Ğ4

4

d. cot x tan x sec x csc x

sin xcos x

cos xsin x

sin2 xcos2 xcos x sin x

1cos x sin x sec x csc x

e. sin x

sec2 x1 1sin2 x

sin x

sin x

tan2 x

sin x cos2 x

sin2 x

cos2 xsin x

1sin2 xsin x

f. tan x 1 2 12 sin x cos x

cos2 x

tan2 x 2 tan x 1

sec2 x 2 tan x

1

cos2 x 2 sin x

cos x 12 sin x cos x

cos2 x


1. f (t) 1.5 sin 2

t 4 2.5 f (t) 1.5 sin 2

t 2 2.5

f (t) 1.5 cos 2

t 5 2.5 f (t) 1.5 cos 2

t 3 2.5

2. f (t) 6 sin 2

t 1 30 f (t) 6 sin 2

t 3 30

f (t) 6 cos 2

t 2 30 f (t) 6 cos 2

t 30

3. f (t) 40 sin 3

t 1.5 40 f (t) 40 sin 3

t 4.5 40

f (t) 40 cos 3

t 3 40 f (t) 40 cos 3

t 40

4. f (t) 13sin 4 t 18 13 f (t) 13sin 4 t 3

8 13

f (t) 13 cos 4 t 14 13 f (t) 13 cos 4 t 13

5. f (t) 3.5 sin 30

t 45 f (t) 3.5 sin 3

t 15 f (t) 3.5 cos

30t f (t) 3.5 cos

3t 30

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 x2 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 x3 1 and f (1) 74

, find f 12 .

Lesson 5.1.2 Transformations of Rational Functions

1. Sketch the graph of f (x) 1x2

1.


2 .

3. Rewrite f (x) 2x5x2

as a transformation of g(x) 1x

and sketch the graph of f (x) .

4. Rewrite f (x) 3x7x3

as a transformation of g(x) 1x

and sketch the graph of f (x) .

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. Given f (x) shown at right, sketch the graph of 1f (x)

.


.


.

x

y f(x)

x

y

x

y f(x)

x

y

x

y

x

y f(x)

4. Give an example of a function f (x) such that 1f (x)

has vertical asymptotes at x 4 and x 3.

5. Give an example of a function f (x) such that 1f (x)

has vertical asymptotes at x 6 and x 5 .

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. limx

5x4 b. lim

x

2x12x1 c. lim

x x

3

d. limx

2x1 e. lim

x3x2 f. lim

xcos x


2 . Use it to evaluate the limit statements below.

a. limx

f (x) b. limx2

f (x) c. limx2

f (x)


1 . Use it to evaluate the limit statements below.

a. limx

f (x) b. limx3

f (x) c. limx3

f (x)

4. Given f (x) 5xx2

, rewrite the function in the form y axh

k . Then evaluate the following limit statements.

a. limx

f (x) b. limx2

f (x) c. limx2

f (x)

5. Given f (x) 3x1x1

, rewrite the function in the form y axh

k . Then evaluate the following limit statements.

a. limx

f (x) b. limx1

f (x) c. limx1

f (x)

Lesson 5.2.3 Working With One-Sided Limits

1. Let f (x) 4x 3x

x.

Complete the table of values below to estimate the value of limx0

4x 3x

x.

x –0.01 –0.001 –0.0001 0.0001 0.001 0.01

f (x)

2. Let f (x) x23x10x2

.


x2 3x10x2

.

x 1.9 1.99 1.999 2.001 2.01 2.1

f (x)

3. Let f (x) x38x2

.


x38x2

.

x 1.9 1.99 1.999 2.001 2.01 2.1

f (x)

4. Use a graph or a table to evaluate the following limits.

a. limx

1x2

5 b. limx4

x2 6x1

c. limx5

2x5

1

5. Use a graph or a table to evaluate the following limits.

a. limx

12x1

3 b. limx2

x2 1x1

c. limx1

3x1

2

Lessons 5.2.4 and 5.2.5 Continuity, Piecewise Functions, and Limits

1. Determine whether the given function is continuous at the point specified and then determine is the limit exists.

Explain your answers.

a. f (x) 3x2 1, x 2 b. f (x) 2x1x4

3, x 4

c. f (x) x2 1 for x 3

2(x 1) for x 3

, x 3 d. f (x) x225x5

1, x 5

2. Find values for m and n such that f (x) will be a continuous function.

a. f (x)

3x m for x 1

2x2 4 for 1 x 1

3x n for x 1

b. f (x)

12

x m for x 4

x2 8 for 4 x 72x n for x 7

3. Given the piecewise function f (x) shown below, evaluate the following expressions.

a. f (6)

b. limx6

f (x)

c. limx3

f (x)

d. f (7)

e. limx7

f (x) f. limx

f (x) g. limx

f (x)

4. Given the piecewise function f (x) shown below, evaluate the following expressions.

a. f (6)

b. limx6

f (x)

c. limx4

f (x)

d. limx4

f (x)

e. f (4) f. limx

f (x) g. limx

f (x)

x

y

2

2

x

y

2

2

5. Let f (x) 2x 1 for x 32x 1 for x 3

.

a. Find limx3

f (x) b. limx0

f (x)

6. Let f (x) x3 2x 1 for x 14x7

x1 for x 1

.

a. Find limx1

f (x) b. limx2

f (x)


1. –4 2. 935

3. 58

4. 4


1. 2.

3. f (x) 1x2

2 4. f (x) 2x3

3

5. f (x) ax2

13 6. f (x) ax9

12

x

y

x

y

x

y

x

y


1. 2.


4. f (x) a(x 4)(x 3)

5. f (x) a(x 6)(x 5)

Lessons 5.2.1 and 5.2.2 Answers

1. 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.

x

y

x

y

x

y


1. 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

2. 7

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

3. 12

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

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 d. 1

e. –3 f. g. –2

4. a. –2 b. 3 c. 1 d. –1

e. –1 f. 1 g.

5. a. 7 b. 0

6. a. DNE b. 13

lesson 13.1.1 circles - chagrin falls exempted village … sem 1 review3.pdf · lesson 13.1.2...

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