blm-9.1 chapter 9 prerequisite skills - anurita dhiman's...

Copyright © 2012, McGraw-Hill Ryerson Limited, ISBN: 978-0-07-073887-4

Name: ___________________________________________ Date: _____________________________

BLM 9–1

Chapter 9 Prerequisite Skills1. Consider the function f(x) x3 2x2 11x 12.

a) Show that x 3 is a factor of f(x). b) If f(x) (x 3)g(x), what is g(x)?

2. Factor each expression fully. a) 30g 4g2 b) 6fg 8g2 c) x2 6x 5 d) 2x2 11x 5 e) 6a2 11a 3 f) x2 0.01 g) 20x2y2 45

3. Consider the sketch of the function h(x) x3 2x2 5x 6.

a) What are the zeros of the function? b) Express h(x) in factored form. c) What are the roots of the equation

x3 2x2 5x 6 0?

4. Identify all non-permissible values for each expression.

a) 43

abc

b) 12 1

xx

c) 2

2 5

2 3 5

x

x x d) 2

1

6 9 t t

e) 6 2(3 1)(2 5)

xx x

f) 1( 1)( 2)( 3) a a a

5. Consider P(x) x3 5x2 2x 8. a) Use the Factor Theorem to find a factor

of P(x). b) Completely factor P(x). c) What is the solution of P(x) 0? d) Sketch the graph of P(x). Explain the

significance of the x-intercepts.

6. a) Arrange the equation 12

4 x

x into the

form ax2 bx c 0, where a, b, and c are integers. State the solution for this equation.

b) Rewrite the function you developed in part a) in the form y (x h)2 k, where h and k are real numbers.

c) Graph the function. Explain the relationship of the x-intercepts and coordinates of the vertex to the functions in parts a) and b).

7. Solve each equation algebraically. Express answers as exact values and identify non-permissible values, where necessary. a) x2 14x 49 0 b) 6x2 17x 3 0 c) x2 6x 7 0 d) 2x2 3x 4 0

e) 2 21

3x x

f) 2

2

4 1 1 4 242 2 4

x x x xx x x

8. Complete the square. Leave your answers in the form a(x h)2 k 0, where a, h, and k are real numbers. a) x2 10x 4 0 b) x2 8x 13 c) 3x2 1 6x d) 2x2 4x 3

9. Solve each equation by graphing the corresponding function.

a) x2 5x 4 0 b) x3 4x2 7x 10 0

10. Simplify each rational expression. State any non-permissible values for the variables.

a) 2

2

1

2 3

x

x x b)

2

2

10 55 75

20 10 150

k kk k

11. Simplify each product or quotient. Identify all non-permissible values.

a) 2

2

4 25 42 13 20 4 10

z zz z z

b) 2 2

2

2 7 15 4 962 10

÷ 3 2

x x xx x

x


Name: ___________________________________________ Date: _____________________________

BLM 9–2

Section 9.1 Extra Practice 1. Match each function with its graph.

a) 43

xy b) 4

3

y

x

c) 43

x

y d) 43 y

x

A

B

C

D

2. Graph the function 5

2

xy using a table of

values. Analyse your graph and use a table to summarize the following characteristics: • non-permissible value(s) • behaviour near non-permissible value(s) • end behaviour • domain • range • equation of vertical asymptote • equation of horizontal asymptote

3. Sketch and graph each function. Identify the domain and range, intercepts, and asymptotes.

a) 3

1

xy

b) 26

xy

c) 54

2x

y

d) 12

8x

y

4. Graph each function using technology, and identify any asymptotes and intercepts.

a) 2 5

1

xx

y

b) 4 3

2

xx

y


Name: ___________________________________________ Date: _____________________________

BLM 9–2 (continued)

5. Write the equation of each function in the

form a

x hy k

.

a)

b)

c)

d)

6. The rational function 5

ax

y k

passes

through points (6, 7) and (4, 1). a) Determine the value of a and k. b) Graph the function.

7. Sketch the graph of 2

1x

y and

2

1

6 9

x xy on the same set of axes.

Describe how one is a transformation of the other.

8. Use a table of values and a graph to analyse

the function .2 1

7

xx

y Then, complete the

table.

Characteristic 2 1

7x

xy

Non-permissible value

Behaviour near non-permissible value

End behaviour

Domain

Range

Equation of vertical asymptote

Equation of horizontal asymptote

9. The distance between two cities is 351 km. a) Write an expression that you can use to

calculate the time, t, in hours, that it takes to travel distance, d, in kilometres, at an average speed of s km/h.

b) Use your formula from part a) to determine how long it will take to travel from one city to the other at an average speed of 65 km/h.

c) If the trip from one city to the other took 5 h, determine the average speed, s, in kilometres per hour.


Name: ___________________________________________ Date: _____________________________

BLM 9–3

Section 9.2 Extra Practice 1. Explain the behaviour at each

non-permissible value in the graph

of the rational function 2

2

5 6

4 21

x x

x xy .

2. Explain how the equation of the rational

function 2

2

3 2

x

x xy can be analysed to

determine whether the graph of the function has an asymptote or a point of discontinuity.

3. Complete the table for the given rational function.

Characteristic ( 3)( 2)( 5)( 3)x xx x

y

Non-permissible value(s)

Feature exhibited at each non-permissible value

Behaviour near each non-permissible value

Domain Range

4. Create a table of values for each function for values near its non-permissible value(s). Explain how your table shows whether a point of discontinuity or an asymptote occurs in each case.

a) 2 5 4

1

x xx

y b) 2

2

5 14

6 8

x x

x xy

5. Analyse each function and predict the location of any vertical asymptotes, points of discontinuity, and intercepts. Then, graph the function to verify your predictions.

a) 2

2

5

7 10

x x

x xy b)

2

2

7 12

9

x x

xy

c) 2 5 4

1

x xx

y d) 22 5 3

3

x xx

y

6. Complete the table and compare the behaviour of the two functions near any non-permissible values.

Characteristic 2 3

3 9x xx

y

2 3

3 9x xx

y




7. Without using technology, match the equation of each rational function with the most appropriate graph. Explain your reasoning.

a) 2

4

3 4

x

x xy b)

2

4

5 4

x

x xy

c) 2 4

4

x xx

y

A


Name: ___________________________________________ Date: _____________________________


B

C

8. Write the equation for each graphed rational function. a)

b)

c)

d)

9. Write the equation of a possible rational function that has an asymptote at x 2, has a point of discontinuity at x 2.5, and passes through (6, 3).


Name: ___________________________________________ Date: _____________________________

BLM 9–4

Section 9.3 Extra Practice 1. Solve each equation algebraically.

a) 2 41 1

5

x x

b) 3 1 35 2 5

xx x

c) 8 6 4 83 6 9

x xx x x

d) 2 2 2 1

2x x

x

2. Solve algebraically. Check your solutions.

a) 139

3

x

x

b) 53

4

xx

x

c) 4 27

4

xx

x

d) 2

23

x

xx

3. a) Determine the roots of the rational

equation 5

6 0 x

x algebraically.

b) Graph the rational function 5

6 x

y x

and determine the x-intercepts. c) Explain the connection between the roots

of the equation and the x-intercepts of the graph of the function.

4. Solve each of the following equations by graphing each side of the equation as a separate function.

a) 62 5

3x

xx

b) 217 3

12 5

x xx

x

c) 22 16

2 13 2

x xx

x

5. Solve by rearranging as a single function and then graphing.

a) 3

4

xx

x

b) 4 21 1

x x

6. Solve each equation algebraically. Give your answers to the nearest hundredth.

a) 4

1

xx

x

b) 21

3

xx

x

c) 35 2

5

x

x

7. Determine the approximate solution(s) to each rational equation graphically, to the nearest hundredth.

a) 2 31 1

3

x x

x xx

b) 3 37

4 9

x

x x

8. Solve the equation 2

18

9 31

n

n n

algebraically.

9. It takes James 9 h longer to construct a fence than it takes Carmen. If they work together, they can construct the fence in 20 h. How long would it take each of them, working alone, to construct the fence?


Name: ___________________________________________ Date: _____________________________

BLM 9–5

Chapter 9 Study Guide This study guide is based on questions from the Chapter 9 Practice Test in the student resource.

Question I can … Help Needed Refer to #1 explain the behaviour of the graph of a rational function

for values of the variable near a non-permissible value some none

9.2 Example 2

#2 match a function to its graph some none

9.2 Example 3

#3 explain the behaviour of the graph of a rational function for values of the variable near a non-permissible value

some none

9.3 Example 2

#4 describe the relationship between the roots of a rational equation and the x-intercepts of the graph of the corresponding rational function

some none

9.3 Example 1

#5 write equations in equivalent forms some none

9.3 Example 2

#6 determine if the graph of a rational function will have an asymptote or a point of discontinuity for a non-permissible value

some none

9.2 Example 2

#7 determine the solution of a rational equation algebraically and graphically

some none

9.3 Example 1

#8 analyse the graph of a rational function to identify characteristics

some none

9.1 Example 1

graph a rational function with linear expressions in the numerator and denominator

some none

9.1 Example 3

#9 determine, graphically, an approximate solution of a rational equation

some none

9.3 Example 2

#10 sketch a graph with or without technology some none

9.2 Example 1

explain the behaviour of the graph of a rational function for values of the variable near a non-permissible value

some none

9.2 Examples 1, 2

#11 analyse a rational function to identify characteristics some none

9.1 Example 3

#12 match a set of rational functions to their graphs, and explain the reasoning

some none

9.2 Example 3

#13 analyse the graphs of a set of rational functions to compare characteristics

some none

9.2 Example 2

#14 solve a rational equation algebraically and graphically some none

9.3 Examples 2, 3

#15 solve problems involving rational equations some none

9.1 Example 5

#16 solve problems involving rational equations some none

9.1 Example 5


Name: ___________________________________________ Date: _____________________________

BLM 9–6

Chapter 9 Test Multiple Choice

For #1 to #8, choose the best answer.

1. The x-intercept of 1

2

kx

y is 0.5. What

is the value of k? A 1.0 B 1.5 C 2.5 D 3.0

2. Consider the function 2

2

1( )

x

xg x . Which

statement is false? A g(x) has two vertical asymptotes. B g(x) is not defined when x 0. C g(x) has one zero. D g(x) is a rational function.

3. Consider the functions f(x) x x2, g(x) 2x 1,

and ( )( )

( ) f xg x

h x . Which statement is true?

A f(x), g(x), and h(x) have the same domain. B The zero of f(x) is the vertical asymptote

of h(x). C The non-permissible value of h(x) is the

zero of g(x). D h(x) is equivalent to y 0.5x 0.25.

4. Consider the following graph of the function 2 1

( )

xx r

f x .

What is the value of r? A 3 B 2 C 2 D 3

5. Which of the following is true of the

rational function 3

26

xy ?

A It has a zero at x 2. B Its range is {yy R}.

C It is equivalent to6 9

2

xx

y .

D It has a vertical asymptote at x 6. 6. The graph of which function has a point of

discontinuity at x 1?

A 2

1

1

xx

y B 2

1

1

xx

y

C 2 1

1

xx

y D 2 1

1

xx

y

7. Which function has a domain of {xx 1, x R} and a range of {yy 3, y R}?

A 31

x

xy B

2

2

3 3

4 3

x xx x

y

C 31

xx

y D 2

2

3

x

x xy

8. How many roots does the equation

2

8 1

16 41

x x have?

A 0 B 1 C 2 D 3

Short Answer 9. a) Sketch the graph of the function

2

2

4

xx

y .

b) Identify the domain, range, and asymptotes of the function.

c) Explain the behaviour of the function as the value of |x| becomes very large.

10. a) Sketch the graph of the function 2

51

xy .

b) State the values of the x-intercept and y-intercept.

c) Solve 2

50 1

x algebraically.

d) How is your answer to part c) related to your answers to parts a) and b)?


Name: ___________________________________________ Date: _____________________________


11. Select the graph that matches the given function.

a) 2

2

( 3)5

xy b)

2

23

( 5)

xy

c) 2

2

( 5)3

xy

A

B

C

12. a) Solve the equation 3

2 17 11

xx

x

algebraically. b) Check your answer to part a) graphically.

Extended Response

13. a) Graph the functions 2

( ) x

f x and

3 11

( )

xx

g x . Use a table to compare

the characteristics of the two graphs. b) Write g(x) as a transformation of f(x):

g(x) f(x a) b. c) Describe the transformation of f(x) to g(x).

14. a) Describe two methods you could use to solve the equation

23 12 54

2(2 1)

x xx

x graphically.

b) Use one of the methods from part a) to solve the equation.

15. A rectangle has an area of 6000 cm2. a) Write an equation to represent length, l,

as a function of the width, w, for this rectangle.

b) Write an equation to represent the change in length, as a function of width, w, when the width is increased by 1 cm.

c) Determine the width, w, of the rectangle if the change in length is 10 cm.

16. An emergency patrol boat is patrolling a river. The river has a 5 km/h current. The patrol boat travels 10 km upriver and 10 km back. The total time, t, in hours, for the round trip is given by the function

2

20

25 v

vt , where v is the speed of the boat

in kilometres per hour. a) State the domain and range for this

function. b) Sketch the graph over the domain

determined in part a). c) Determine the speed of the boat if the

round trip took 1.5 h.


BLM 9–7

Chapter 9 BLM Answers BLM 9–1 Prerequisite Skills 1. a)

2

3 2

3 2

2

2

5 4

3 2 11 12

3

5 11

5 15

4 12

4 12

0

x xx x x x

x x

x x

x xxx

b) g(x) x2 5x 4 2. a) 2g(15 2g) b) 2g(3f 4g) c) (x 5)(x 1) d) (2x 1)(x 5) e) (2a 3)(3a 1) f) (x 0.1)(x 0.1) g) 5(2xy 3)(2xy 3) 3. a) 2, 1, and 3 b) h(x) (x 2)(x 1)(x 3) c) 2, 1, and 3

4. a) b 0, c 0 b) 1

2x c) 5

2x , x 1 d) t 3

e) 1 5

3 2, x x f) a 1, a 2, and a 3

5. a) P(1) (1)3 5(1)2 2(1) 8 0, therefore, (x 1) is a factor. b) P(x) (x 1)(x 4)(x 2) c) x 1, 2, 4 d)

The x-intercepts are the solution to P(x) 0. 6. a) x2 4x 12 0; 6 and 2 b) y (x 2)2 16 c)

The x-intercepts are the solution to the equation. The x-coordinate of the vertex is the value that makes (x h) 0 and the y-coordinate is equal to k.

7. a) x 7 b) 1

63, x c) 3 2 x

d) 3 41

4

x e) 2

3 x , x 0, x 1

f) x 6, x 2, 2 8. a) (x 5)2 21 0 b) (x 4)2 3 0 c) 3(x 1)2 2 0 d) 2(x 1)2 5 0 9. a) x 4, 1

b) x 5, 1, 2

10. a) 1

3

xx

, x 1, 3 b) 3 5

2 3 2, , 3

kk

k

11. a) 1 5

2 2, 4, z b) 3 3

2, , 0,5

x

x

BLM 9–2 Section 9.1 Extra Practice 1. a) B b) C c) D d) A 2.



Characteristic 5

2xy

Non-permissible value x 2


As x approaches 2, |y| becomes very large.

End behaviour As |x| becomes very large, y approaches 0.

Domain {xx 2, x R}

Range {yy 0, y R}


x 2


y 0

3. a)

domain: {xx 1, x R}; range: {yy 0, y R}; intercept: (0, 3); asymptotes: x 1, y 0

b)

domain: {xx 0, x R}; range: {yy 6, y R};

intercept: 1

3, 0 ; asymptotes: x 0, y 6

c)

domain: {xx 4, x R}; range: {yy 2, y R}; intercepts: (0, 0.75), (1.5, 0); asymptotes: x 4, y 2

d)

domain: {xx 2, x R}; range: {yy 8, y R}; intercepts: (0, 8.5), (2.125, 0); asymptotes: x 2, y 8

4. a)

asymptotes: x 1, y 2; intercepts: (2.5, 0), (0, 5) b)

asymptotes: x 2, y 4; intercepts: (0, 1.5), (0.75, 0)

5. a) 3x

y b) 4x

y c) 2

5

xy d) 2

4

xy

6. a) a 3, k 4 b)



7.

The graph of 21

6 9

x xy is the graph of 2

1x

y

translated 3 units left.

8. x y 5 0.92 2 0.56

1 0.17 4 2.33 7 undefined

10 6.33 13 4.17 16 3.44 19 3.08

Characteristic 2 1

7x

xy

Non-permissible value x 7



End behaviour As |x| becomes very large, y approaches 2.

Domain {xx 7, x R}

Range {yy 2, y R}


x 7


y 2

9. a) ds

t

b) 351

655.4 t , so 5.4 hours or 5 h and 24 min

c) 70.2 km/h BLM 9–3 Section 9.2 Extra Practice 1. point of discontinuity at (3,

1

10) vertical

asymptote: x 7

2. You can factor the denominator: 2

( 2)( 1)

xx x

y .

Since the factor (x 2) appears in the numerator and denominator, the graph will have a point of discontinuity at (2, 1). The factor (x 1) appears in the denominator only, so there will be an asymptote at x 1.

3.

Characteristic

( 3)( 2)( 5)( 3)x xx x

y

Non-permissible value(s) x 5 and x 3


asymptote at x 5; point of discontinuity at (3, 2.5)


As x approaches 5, |y|

becomes very large.

As x approaches 3, y approaches 2.5.

Domain {xx 3, 5, x R}

Range {yy 1,

5

2, y R}

4. a) x y

0.9 3.1 0.99 3.01 0.999 3.001 0.9999 3.0001 1 undefined 1.0001 2.9999 1.001 2.999 1.01 2.99 1.1 2.9

As x approaches 1, y approaches 3.



b) x y

1.9 4.238 095 24 1.99 4.472 636 82 1.999 4.497 251 37 1.9999 4.499 725 01 2 undefined 2.0001 4.500 275 01 2.001 4.502 751 38 2.01 4.527 638 19 2.1 4.789 473 68

x y

3.9 109 3.99 1 099 3.999 10 999 3.9999 109 999 4 undefined 4.0001 110 001 4.001 11 001 4.01 1 101 4.1 111

As x approaches 2, y approaches 4.5, and as x approaches 4, |y| becomes very large, approaching negative infinity or positive infinity.

5. a) vertical asymptote: x 2; point of

discontinuity at (5, 5

3);

x-intercept: (0, 0); y-intercept: (0, 0)

b) vertical asymptote: x 3; point of discontinuity

at (3, 1

16 );

x-intercept: (4, 0); y-intercept: (0, 4

3 )

c) no vertical asymptote; point of discontinuity at (1, 3); x-intercept: (4, 0); y-intercept: (0, 4)

d) no vertical asymptote; point of discontinuity at (3, 7); x-intercept: (0.5, 0); y-intercept: (0, 1)

6.

Characteristic 2 3

3 9x x

xy

2 33 9

x xx

y


x 3 x 3


point of discontinuity

asymptote


As x approaches 3, y approaches 1.


7. a) C; Example: In factored form, the rational function has two non-permissible values in the denominator, which do not appear in the numerator. Therefore, the graph with two asymptotes is the most appropriate choice. b) B; Example: In factored form, the rational function has one non-permissible value that appears in both the numerator and denominator, and another non-permissible value that is only in the denominator. Therefore, the graph with one asymptote and one point of discontinuity is the most appropriate choice. c) A; Example: In factored form, one non-permissible value appears in the numerator and denominator. Therefore, the graph has a point of discontinuity, but no asymptote.



8. a) 3 2

3

x x

xy or

2 6

3

x xx

y

b)

2 2

2

x x

xy or

2 4

2

xx

y

c)

4

4 4

xx x

y or 2

4

16

xx

y

d)

5

3 5

x

x xy or 2

5

8 15

xx x

y

9. Example:12(2 5)

( 2)(2 5)

xx x

y

BLM 9–4 Section 9.3 Extra Practice 1. a) 3

5x b) x 5 c) x 24 d) x 4

2. a) x 10 and x 4 b) x 7 and x 1

c) x 10 and x 3 d) 3

2x and x 2

3. a) x 5 and x 1

b)

c) The value of the function is 0 when the value of x is 1 or 5. The x-intercepts of the graph of the function are the same as the roots of the corresponding equation. 4. a) x 0 and x 3.5

b) x 2 and x 6

c) x 0.25 and x 2

5. a) 0 x2 8x 12

x 2 and x 6



b) 2

6 2

1

xyx

x 3 6. a) x 0.76 and x 5.24 b) x 2.79 and x 1.79 c) x 0.53 and x 4.87 7. a)

x 0.63 b)

x 0.85 and x 6.15 8. The solution n 3 is a non-permissible value, so there is no solution. 9. Carmen: 36 h; James: 45 h

BLM 9–6 Chapter 9 Test 1. D 2. B 3. C 4. D 5. C 6. A 7. C 8. B 9. a)

b) domain: {xx 2, x R};

range: {yy 0, 1

4 , y R};

vertical asymptote: x 2; horizontal asymptote: y 0 c) As |x| becomes very large, y approaches 0.

10. a)

b) x-intercept 3; y-intercept 0.6 c) x 3 d) The root of the equation is the same as the x-intercept. 11. a) B b) A c) C 12. a) x 1 b)



13. a)

Characteristic 2( )x

f x 3 11

( ) xx

g x

Non-permissible value

x 0 x 1




End behaviour

As |x| becomes very large, y approaches 0.

As |x| becomes very large, y approaches 3.

Domain {x x ≠ 0, x R}

{x x ≠ 1, x R}

Range {y y ≠ 0, y R}

{y y ≠ 3, y R}


x 0 x 1


y 0 y 3

b) g(x) = f (x 1) + 3 2

1( ) 3

xg x

c) a vertical translation 3 units up and a horizontal translation 1 unit right 14. a) Example:

Method 1: Graph 23 12 5

42(2 1)

x xx

y x , and

determine the x-intercepts of the graph.

Method 2: Graph 23 12 5

4

x xx

y and y 2(2x 1),

and determine the x-coordinates of the points where the two graphs intersect. b)

x 1 and x 3

15. a) 6000

wl

b)

2

6000 6000

1

6000

change in

w w

w w

l

c) w 24 cm 16. a) domain: {vv > 5, v R}; range: {tt > 0, t R} b)

c) v 15 km/h

blm-9.1 chapter 9 prerequisite skills - anurita dhiman's...

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