grade 12 pre-calculus mathematics notebook chapter 9

Grade 12

Pre-Calculus Mathematics Notebook

Chapter 9

Rational

Functions

Outcomes: R14

MPC40S Date:_______________

Chapter 9 – Homework

Section Page Questions

MPC40S Date:_______________

Chapter 9: RATIONAL FUNCTIONS

9.1 – Exploring Rational Functions: khx

ay +

−=

Sketch the graph of ( )x

xf1

= .

Note, x = 0 is a ___________________ Graphically this creates a ___________________ When x → ∞, y → and when x → -∞, y → Thus, y = 0 is a _______________ The curves are in Quadrants ______

Sketch the graph of ( )1

f xx

= − .

The curves are now in Quadrants _______

Note: 1

x− is the same as

1

x

−.

x y

R14

Given ( )x

xf1

= , we can sketch the graph of ( ) 123 −+= xfy .

Note:

The equation of the

transformed graph is

12

3−

+=

xy .

Can you see the

connection?

The general equation of a rational function is khx

ay +

−= .

This represents a vertical stretch by a factor of a, followed by a horizontal shift of h units, and a vertical shift of k units.

x=h is a _______________________, y= k is a ______________________

Explain the behaviour of the graph for values of the variable around x = -2.

______________________________________________________________________

Explain the end behaviour of the graph.

MPC40S Date:_______________

Example #1

Sketch the graph of 1

13

yx

= +−

Note : You must label a point in each section of the graph

non-permissible value

x-intercept

y-intercept

vertical asymptote

horizontal asymptote

domain

range

Example #2


2+

−

−=

xy .


x-intercept

y-intercept

vertical asymptote


domain

range

MPC40S Date:_______________

Example #3


1

xxf = .

Example #4


1

xxf −= .

x y

x y

Note: We can use the previous ideas to help us graph the transformed versions of these functions.

9.1 Homework Assignment: Graph each of the following functions

1) 2)

3) 4)

MPC40S Date:_______________

5) 6)

7)

State

a) Domain b) Range c) Vertical Asymptote d) Horizontal Asymptote

MPC40S Date:_______________


9.2: Graphing Rational Functions of the form

Example #1


43

−

−=

x

xy .

T: Determine the Vertical Asymptote(s) Asymptotes will occur when g(x) = 0 (note: these are also NPV’s) II: Determine the Horizontal Asymptote

• If the degree of the numerator equals the degree of the denominator, then the equation of the horizontal asymptote is “y = ratio of leading coefficients”.

Ex: 86

432

2

−

+=

x

xy eqn of h.a.: _____

5

273

32

−

−+−=

x

xxxy eqn of h.a.: _____

• If the degree of the numerator is less than the degree of the denominator,

then the horizontal asymptote is 0=y .

Ex: 1

12 +

=x

y eqn of h.a.: ______ 5

13 +

−=

x

xy eqn of h.a.: ______

• If the degree of the numerator is greater than the degree of the denominator, then there is no horizontal asymptote.

Ex: 3

9+=


1

12

4

+

−=

x


III: Determine at least one point in each section and use your knowledge of asymptotic behavior to construct the graph.

R14

Example 1 continued:


43

−

−=

x

xy .

vertical asymptote


domain

range

MPC40S Date:_______________

Example #2

Sketch the graph of 42 −

=x

xy .

vertical asymptote(s)


domain

range

Example #3


12 +

=x

y .


x-intercept

y-intercept

vertical asymptote


domain

range

MPC40S Date:_______________

Example #4

Ex4. Sketch the graph of ( )32

12 −−

+=

xx

xxf .

Remember to first factor the numerator and denominator if possible. A ____________________ is a point at which the graph of a function is not continuous. This point is missing from the graph and so we represent it with an open circle.

domain

range

Example #5

Sketch the following:

2

652

−

+−=

x

xxy

MPC40S Date:_______________

Example #6

What type of discontinuity will this function have?

23

22 +−

−=

xx

xy

Graph the function.

9.2 Homework Graphing rational functions – R14

1.

2.

MPC40S Date:_______________

3.

4.

MPC40S Date:_______________


9.3 – Connecting Graphs and Rational Equations

Example #1

Ex1: Solve the following equations algebraically and graphically.

a) Algebraically Graphically

R14

b) 21

2−=

−x

x

Algebraically Graphically y1 = y2 =

MPC40S Date:_______________

Chapter 9 Review

1. Explain the difference between a point of discontinuity and an asymptote. Next, give two

examples of equations, one with only a point of discontinuity and one with only an

asymptote.

2. a) Sketch the graph of the function ( )2

1

2 3

xf x

x x

−=

+ −.

b) State the domain and range of this function.

3. Sketch the graph of the function ( )1

3f x

x= −

+.

4. Sketch the graph of the function ( )2 4

1

xf x

x

−=

−.

MPC40S Date:_______________

5. Sketch the graph of the function ( )( )( )

2

3 1f x

x x=

− +.

6. Sketch the graph of the function ( )( )( )3 1

xf x

x x=

− +.

7. a) Sketch the graph of the function 2

622

2

−+

−+=

xx

xxy .

b) Explain how you can find the equation of the horizontal asymptote without having to

divide the two functions. Next, explain why this will not always work.

8. a) Sketch the graph of the function 1

22 +

−=

xy .

b) Write the equations of all asymptotes for

this function.

MPC40S Date:_______________

9. What is the x-intercept of ?

A. There is no x-intercept. C. −2 B.

−1

2

D. 0

10. Which function has vertical asymptotes with equations and −6

7?

A.

C.

B.

D.

11. Solve the equation .

A.

C. no solution

B. x = 2, x = 9 D. 3 2

12. Which function has a graph in the shape of a parabola?

A. C.

B. D. none of the above

13. What are the x-intercepts of the graph of ?

A. –7, –5 C. 7, 5

B. 2, –9 D. –2, 9

14. Explain what happens to the graph of as it approaches x=1 from the left hand

side. Explain what happens to the graph as it approaches x=1 from the right hand side.

15. Match the equation of each rational function with the most appropriate graph. Explain your reasoning.

16. Write the eqatuion for each rational function below.

a) c)

b) d)

MPC40S Date:_______________

ANSWERS

MPC40S Date:_______________

grade 12 pre-calculus mathematics notebook chapter 9

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