grade 12 pre-calculus mathematics notebook chapter 9
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Page 1
Grade 12
Pre-Calculus Mathematics Notebook
Chapter 9
Rational
Functions
Outcomes: R14
MPC40S Date:_______________
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Page 3
Chapter 9 – Homework
Section Page Questions
MPC40S Date:_______________
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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
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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:_______________
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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
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Example #2
Sketch the graph of 31
2+
−
−=
xy .
non-permissible value
x-intercept
y-intercept
vertical asymptote
horizontal asymptote
domain
range
MPC40S Date:_______________
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Example #3
Sketch the graph of ( )2
1
xxf = .
Example #4
Sketch the graph of ( )2
1
xxf −= .
x y
x y
Note: We can use the previous ideas to help us graph the transformed versions of these functions.
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9.1 Homework Assignment: Graph each of the following functions
1) 2)
3) 4)
MPC40S Date:_______________
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5) 6)
7)
State
a) Domain b) Range c) Vertical Asymptote d) Horizontal Asymptote
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MPC40S Date:_______________
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Chapter 9: RATIONAL FUNCTIONS
9.2: Graphing Rational Functions of the form
Example #1
Sketch the graph of 2
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+=
xy eqn of h.a.: ______
1
12
4
+
−=
x
xy eqn of h.a.: ______
III: Determine at least one point in each section and use your knowledge of asymptotic behavior to construct the graph.
R14
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Example 1 continued:
Sketch the graph of 2
43
−
−=
x
xy .
vertical asymptote
horizontal asymptote
domain
range
MPC40S Date:_______________
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Example #2
Sketch the graph of 42 −
=x
xy .
vertical asymptote(s)
horizontal asymptote
domain
range
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Example #3
Sketch the graph of 1
12 +
=x
y .
non-permissible value
x-intercept
y-intercept
vertical asymptote
horizontal asymptote
domain
range
MPC40S Date:_______________
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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
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Example #5
Sketch the following:
2
652
−
+−=
x
xxy
MPC40S Date:_______________
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Example #6
What type of discontinuity will this function have?
23
22 +−
−=
xx
xy
Graph the function.
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9.2 Homework Graphing rational functions – R14
1.
2.
MPC40S Date:_______________
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3.
4.
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MPC40S Date:_______________
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Chapter 9: RATIONAL FUNCTIONS
9.3 – Connecting Graphs and Rational Equations
Example #1
Ex1: Solve the following equations algebraically and graphically.
a) Algebraically Graphically
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b) 21
2−=
−x
x
Algebraically Graphically y1 = y2 =
MPC40S Date:_______________
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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.
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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:_______________
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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=
− +.
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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:_______________
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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.
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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:_______________
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ANSWERS
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MPC40S Date:_______________
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