9.1 Exploring Rational Functions using Transformations (Day 1)

A rational function is a function that is formed by the ratio of two polynomials. It is written in the form

p xf x

q x where p x and q x are polynomials and 0q x .

Why does the definition of a rational function specify that 0q x ?

What will this influence?

What does the graph of a rational function look like? Consider the function 1

f xx

.

A table of values can be used to graph the function for the first time.

x -

100 -50 -10 -1 -0.5 -0.1 0 0.1 0.5 1 10 50 100

f(x) 0.01 0.02 0.1 1 2 100 Undef 100 2 1 0.1 0.2 0.01

What happens to the graph as x approaches 0?

- From the left,

- From the right,

What happens to the graph as y approaches 0?

- From above,

- From below,

An asymptote is a line that the graph of a relation approaches as a limit or boundary.

The graph of a rational function never crosses a vertical asymptote but it may or may not cross a horizontal asymptote.

At what values do vertical asymptotes occur?

Equation of the vertical asymptote of 1

f xx

: Equation of the horizontal asymptote of 1

f xx

:

2

Consider the graphs of the following functions:

How did adding in the ‘4’ to the equation of the four functions transform the graph of 1

f xx

?

In Chapter 1, you were introduced to three types of transformations which can alter the equation or graph of

a function.

A translation results in a change of _________________ while preserving the shape and orientation.

A reflection may result in a change of _________________while preserving the shape.

A stretch results in a change of _________________while preserving the orientation.

4y

x 1

4y

x

1

4y

x

1

4y

x

1

4y

x

3

Given base function or parent function y f x , multiple transformations can be applied using the general

transformation model y a f b x h k . The mapping notation for multiple transformations would be:

, ___________, ___________x y

Consider the base function will be the rational function 1

f xx

. What are the characteristics of the base

function?

non-permissible value(s)

behaviour near non-permissible value(s)

end behaviour

domain

range

equation of vertical asymptote

equation of horizontal asymptote

Example 1:

Sketch the graph of the function 𝑦 =6

𝑥−2− 3 using transformations, and identify any important characteristics

of the graph.

4

Example 2: Sketch the graph of the function 4

12

yx

by transforming the graph of 1

yx

.

Example 3: Write the equation for the function in the form

ay k

b x h

.

Assignment Pg 442 #1, 3, 7abd

5

9.1 Exploring Rational Functions using Transformations (Day 2)

We will also use 2

1f x

x as a base function. A table of values can be used to graph the function for the

first time.

x -100 -50 -10 -1 -0.5 -0.1 0 0.1 0.5 1 10 50 100

f(x) -0.00001 -

0.0004 -0.01 -1 4 100 Undef 100 4 1 0.01 0.0004 0.00001

non-permissible value(s)

behaviour near non-permissible value(s)

end behaviour

domain

range

equation of vertical asymptote

equation of horizontal asymptote

Example 1: Sketch the graph of the function 2

42

6 9y

x x

by transforming the graph of

2

1y

x .

Assignment Pg 442 #2, 6, 12

6

Analyzing Functions not written in the Form

ay k

b x h

or

2

ay k

b x h

:

Consider the graph of 2 2

4

xy

x

.

Identify any asymptotes and intercepts.

How does this graph to relate to the graph of 1

yx

?

We can use synthetic division to manipulate the equation of 2 2

4

xy

x

so that it is written in the form

a

y kb x h

.

We look at another approach to graphing rational functions in section 9.2, when the function is not given in

the form

ay k

b x h

or

2

ay k

b x h

.

Assignment Pg 442 #2, 5 (don’t have technology)

7

9.2 Analyzing Rational Functions (Day 1)

Consider the function 2 2

2

x xy

x

. What value of x is important to consider when analyzing this

function? Is this the graph you anticipated?

How does the behavior of the functions near its non-

permissible value differ from the rational functions

you looked at previously?

Now, factor the function and reduce.

Does the reduced form show the reason why the graph looks as it does?

Graphs of rational functions can have a variety of shapes and different features – vertical asymptotes and

points of discontinuity (looks like a hole) are possible.

A vertical asymptote occurs where a non-permissible value exists (in the denominator).

A point of discontinuity occurs when the function can be simplified by dividing the numerator and

denominator by a common factor that includes a variable. The factor that is eventually reduced is the

one that creates the non-permissible value that is the point of discontinuity.

Example 1: Sketch the graph of 2 5 6

3

x xf x

x

. Analyze its behavior near its non-permissible value.

(Factor, reduce, etc.)

To find the coordinate of the point of discontinuity, substitute into the simplified form of the function!

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-8

-6

-4

-2

2

4

6

8

x

y

8

Points of Discontinuity vs Asymyptotes:

The non-permissible value for both functions is 2. However, the graph on the left does not exist at (2, –1), whereas the the graph on the right is undefined at x = 2.

x- and y-Intercepts:

Do you remember how to find an x-intercept?

For 2 5 6

3

x xf x

x

,

Do you remember how to find a y-intercept?

For 2 5 6

3

x xf x

x

,

Assignment Pg 451 #1, 4(predictions, no graphs)

9.2 Analyzing Rational Functions (Day 1)

Consider the function 2 2

2

x xy

x

. What value of x is important to consider when analyzing this

function? Is this the graph you anticipated?

How does the behavior of the functions near its non-

permissible value differ from the rational functions

you looked at previously?

Now, factor the function and reduce.

Does the reduced form show the reason why the graph looks as it does?

Graphs of rational functions can have a variety of shapes and different features – vertical asymptotes and

points of discontinuity (looks like a hole) are possible.

A vertical asymptote occurs where a non-permissible value exists (in the denominator).

A point of discontinuity occurs when the function can be simplified by dividing the numerator and

denominator by a common factor that includes a variable. The factor that is eventually reduced is the

one that creates the non-permissible value that is the point of discontinuity.

Example 1: Sketch the graph of 2 5 6

3

x xf x

x

. Analyze its behavior near its non-permissible value.

(Factor, reduce, etc.)

To find the coordinate of the point of discontinuity, substitute into the simplified form of the function!

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-8

-6

-4

-2

2

4

6

8

x

y

10

Points of Discontinuity vs Asymyptotes:

The non-permissible value for both functions is 2. However, the graph on the left does not exist at (2, –1), whereas the the graph on the right is undefined at x = 2.

x- and y-Intercepts:

Do you remember how to find an x-intercept?

For 2 5 6

3

x xf x

x

,

Do you remember how to find a y-intercept?

For 2 5 6

3

x xf x

x

,

Assignment Pg 451 #1, 4(predictions, no graphs)

11

9.2 Analyzing Rational Functions (Day 2)

Ex. Graph 2

4

5 4

xy

x x

. (First, factor and state the non-permissible values.)

x-intercept(s) y-intercept(s)

• Occur when y = 0 • Occur when x = 0

Can this really happen in this case?

Vertical asymptote(s) or Point(s) of discontinuity

Occur where the function is undefined; where the denominator = 0.

Horizontal asymptote(s)

1. If the numerator and the denominator have the same degree, then the equation of the horizontal

asymptote line is “y = ratio of leading coefficients”.

2. If the degree of the numerator is less than the degree of the denominator, then the equation of the

horizontal asymptote is “y = 0”.

These are the only cases we will experience.

Now to sketch the graph…

Place asymptotes

Place x- and y-intercepts

Move according to sign analysis

Use asymptotes as ‘framework’

12

Examples:

Write the equation for the rational functions shown below.

a)

Characteristics Implications for Factors

b) Characteristics Implications for Factors

Assignment Pg 451 #4abc (graphs), 5 - 8

13

9.3 Connecting Graphs and Rational Equations

Rational equations can be solved algebraically or graphically.

When solving algebraically, watch for extraneous roots.

Example 1: Solve 16

46

xx

graphically using technology.

Note that the LHS of this equation is a rational function and the RHS is linear.

Solving Algebraically

Factor numerator and denominator fully. Note any restrictions on the variable.

Multiply each term in the equation by the lowest common denominator (LCD) in order to reduce the

denominators. You should no longer have denominators after this step.

Solve for x.

Check for extraneous roots (use the original equation). Does the solution have any non-permissible

values?

Example 2: Solve 16

46

xx

algebraically.

14

Example 3: Solve 8 15

22 5 4 10

x xx

x x

Assignment Pg 465 #3, 6 (algebraically only)