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Mary Ward C.S.S. Mathematics Department MHF 4U1 Advanced Functions Unit 10 RATIONAL FUNCTIONS Rational Functions ‘Donut Holes’ Asymptotes Inverse Proportionality: Continuity of a Function. How to Distinguish Vertical Asymptotes From Donut Holes Linear Rational Function: Linear Rational Function as Transformation of The End Behaviour of a Rational Function Curve Sketching February, 2011

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Page 1: MHF 4U Unit 10 Rational Functions New

Mary Ward C.S.S.

Mathematics Department

MHF 4U1

Advanced Functions

Unit 10

RATIONAL FUNCTIONS

Rational Functions ‘Donut Holes’ Asymptotes

Inverse Proportionality:

Continuity of a Function. How to Distinguish Vertical Asymptotes From Donut Holes

Linear Rational Function:

Linear Rational Function as Transformation of

The End Behaviour of a Rational Function Curve Sketching

RATIONAL FUNCTIONS

February, 2011

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MHF 4U Unit 10: RATIONAL FUNCTIONS

EXPECTATIONS: By the end of this unit, students will:

determine, from the equation of a rational function, the intercepts and the positions of the vertical and the horizontal or oblique asymptotes to the graph of the function;

determine, through investigation with and without technology, key features of the graphs of rational functions that are the reciprocals of linear and quadratic functions and make connections between the algebraic and graphical representations of these rational functions;

determine, through investigation with and without technology, key features of the graphs of rational functions that have linear expressions in the numerator and denominator, and make connections between the algebraic and graphical representations of these rational functions;

sketch the graph of a simple rational function using the key features, given the algebraic representation of the function.

SPECIAL INSTRUCTIONS: a graphing calculator is mandatory for this unit.

TEXT REFERENCE: Use this learning guide and Advanced Functions textbook, study the examples Chapter 5 Section 5.1, 5.2, 5.3. Upon the unit completion it is strongly recommended to do the Chapter 5 Self-Test corresponding questions.

ASSESSMENT / EVALUATION

ACTIVITY # NAME OF ACTIVITYKICA CATEGORY

DESCRIPTION (QUIZ, TEST, INVESTIGATION, PROBLEM)

TIME

A Rational and Reciprocal Functions

K/C Problems 1h

B Exploring Quotients of Polynomial Functions

T/C Problems 1h

C Graphs of Rational Functions

K/C Problems 0.5h

D The End Behavior of a Rational Function

T/C Problems 0.5h

E Summative A/C Seminar Quiz and Unit 10 Test

2h

Total Estimated Time: 5 h

2

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MHF 4U Unit 10: RATIONAL FUNCTIONS

RATIONAL FUNCTIONS

Rational function is formed when a polynomial is divided by a polynomial.

This is a function of the form

,

where and are both polynomials and is not zero.

Donut Holes

Example 1 Consider the function

Obviously, f is defined for all real numbers except .

We cannot divide by zero. When x=0, the function is undefined:

.

For all other values of x we can cancel x in the numerator and denominator. This will simplify the function to

for

This would be the equation of the straight line with a hole at x = 0.

The point (0, 2) does not exist, but leaves a “donut hole” on the line.

The domain is , .The range is .

Example 2 Consider the function

3

The graph of a rational function may have donut holes or asymptotes or both depending on the denominator.

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MHF 4U Unit 10: RATIONAL FUNCTIONS

.

This function is defined for all real numbers except 2.

The polynomials in the numerator and the denominator of the above function would factor like this:

.

For any we can cancel the common factor, getting

for .

The graph of is that of the straightline with one point removed,namely the point (2, 4).

At this point there is a ‘donut hole’.

The domain is , .

The range is , .

Asymptotes

An asymptote of a function is a line to which the graph of the function approaches such that it never crosses or touches this line.

An assymptote parallel to the x-axis is called a horozontal asymptotes.

Aa assymptote parallel to the y-axis is a vertical asymptotes.

If an asymptote is not parallel to x- or y-axis, it is called an oblique asymptote or, sinonimously, a slant asymptote.

Inverse proportionality:

4

To understand the behaviour of a rational function it is useful to factor the polynomials (if possible).

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MHF 4U Unit 10: RATIONAL FUNCTIONS

Let us start with the simplest rational function

,

The function is the reciprocal of the function . As x approaches

zero, approaches positive or negative infinity, as it is shown in the

figure on the right.

As x approaches positive or negative

infinity, approaches zero.

The graph is a hyperbola with the

coordinate axes as the asymptotes:

Horizontal Asymptote: the x-axis (y=0).

Vertical Asymptote: the y-axis (x=0).

The graph of is obtained from the graph of by stretching or

compressing it vertically and reflecting it in the x-axis in the case where is negative.

The two graphs below illustrate these transformations.

RECIPROCAL OF A QUADRATIC FUNCTION

Example 3

Graph the function

5

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MHF 4U Unit 10: RATIONAL FUNCTIONS

.

Solution:

1. Factor the denominator of and represent the function in factored form:

Then

2. domain f: , , ,

3. The sign of the function is the same as the sign of the polynomial in the denominator. Graph the function .

is positive on , and ,

is negative on ,

4. has vertical asymptotes at

and

5. To find the x-intercept, set y equal to zero:

.

Since the numerator of is not zero, y cannot be zero. Therefore there is no y-intercept. To find the y-intercept, set x equal to zero.

y-intercept:

6. The quadratic trinomial in the denominator has a minimum value ymin= -4 at x=4 . This corresponds to the local maximum:

6

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MHF 4U Unit 10: RATIONAL FUNCTIONS

local maximum: .

7. As x approaches positive or negative infinity, approaches zero.

Horizontal asymptote: y=0 (the x-axis).

8. The function is neither even nor odd.

The graph is not symmetric about

the y-axis or about the origin.

Here is how the function looks on the graph.

pg. 255 # 5 a), d), f), g), pg. 256 # 6, 8 b), c), f), pg. 257 # 12, 15

Continuity of a FunctionHow to Distinguish Vertical Asymptotes from Donut Holes

The formal definition of continuity is based on the notion of limit. It will be given later in the course. An intuitive definition is as follows.

Functions whose graph has vertical asymptotes and/or donut holes are discontinuous.

Both vertical asymptotes and donut holes correspond to the zeroes of

the denominator of a rational function . They occur at the

7

Assignment A: RATIONAL AND RECIPROCAL FUNCTIONS

A function whose graph can be drawn without lifting the pencil from the paper is a continuous function.

Otherwise the function is discontinuous.

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MHF 4U Unit 10: RATIONAL FUNCTIONS

points where . These points are called the singularities of the rational function.

Recall that if , then is a factor of . (The Factor Theorem)

In the most general case, may be a multiple factor of both and , like in

,

where and are polynomials.

Then, if , there will be a donut hole in the graph at .

If , then there will be an asymptote at .

For example, the graph of the function

is the same as the graph of the function

for .

with a donut hole at .

On the other hand, the function

is identical to the function

and its graph has a vertical asymptote at .

And, finally, not any rational function is discontinuous. Some of them are continuous.

For example, the denominator of the rational function

8

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MHF 4U Unit 10: RATIONAL FUNCTIONS

is always positive. Therefore the function is continuous. It is defined for any x and its graph does not have any vertical asymptotes or donut holes.

pg. 262 # 2 a), c), d), j), h), 3

LINEAR RATIONAL FUNCTION

This function is defined for any x except .

What happens at , depends on , , , and .

If and , then the function is simplified to

, for .

In this case, the graph is the graph of the line with a donut hole at

. If the polynomials in the numerator and denominator are

different, the function will have a vertical asymptote at .

Now we shall concentrate on the horizontal asymptotes.

In the function , the leading term of is ; the

leading term of is . As x approaches positive or negative infinity , the function starts acting like

.

9

Assignment B: EXPLORING QUOTIENTS OF POLYNOMIAL FUNCTIONS

To find the horizontal asymptotes, consider the end behaviour of the function.

The end behaviour describes how the function behaves as x approaches positive and negative infinity.

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MHF 4U Unit 10: RATIONAL FUNCTIONS

Thus, as or , y will approach the horizontal line .

By definition, this line is a horizontal asymptote.

Example 4 Consider the function

.

The function is not defined at x=2.5 where the graph has a vertical asymptote.

Vertical Asymptote:

Consider the end behaviour of .

The leading term of 6x-1 is 6x; the leading term of 2x-5 is 2x.

As or , these terms becomes dominant and the function will behave like

, or, equivalently, .

Then the left and the right edges of the graph will approach the horizontal asymptote .

Horizontal Asymptote:

To find the x-intercepts, set y = 0:

, , .

Thus the x-intercept is .

To find the y-intercepts, set x = 0:

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To sketch the graph of a function, it is useful to find its x- and y-intercepts and the intervals on which the function is positive or negative.

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MHF 4U Unit 10: RATIONAL FUNCTIONS

Thus the y-intercept is .

The numerator, , changes sign at ; the denominator, ,

changes sign at . These values divide the x-axis into three intervals:

, , and .

For x > 2.5 the function is positive. Then it changes sign at x= 2.5 and

, as

shown in the diagram above.

One can see that is positive for and negative

for .

Now we are ready to sketch the graph.

To do this, realize the following:

the function has a horizontal asymptote ; it has a vertical asymptote ;

the function crosses the x-axis at ;

the function crosses the y-axis at ;

the function is positive for and negative for

.

We mark the intercepts and draw the asymptotes.

11

An asymptote is not a part of the graph. Draw asymptotes as dashed lines.

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MHF 4U Unit 10: RATIONAL FUNCTIONS

Here is the graph of the function.

Linear Rational Function as Transformation of

The graph of looks like the graph of translated

horizontally and vertically.

To see that this is so, consider the function .

Divide the polynomial in the numerator by the polynomial in the denominator.

Write this as a division statement

Dividing the numerator and denominator by 2, we get

Thus

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MHF 4U Unit 10: RATIONAL FUNCTIONS

can be written as

.

Observe that the graph of this function is the graph of translated

2.5 units to the right and 3 units up.

In general, the graph of any linear rational function can be

obtained by translating the graph of .

The End Behaviour of a Rational Function

Consider a rational function

.

The leading term of is .

The leading term of is .

As takes on large positive or large negative values , the function will behave like

.

Depending on and , we have the following end behaviour.

If , and , the function behaves like , where

. As , . The graph has the x-axis as a horizontal linear asymptote.

If and , then y will approach a constant value of .

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MHF 4U Unit 10: RATIONAL FUNCTIONS

The graph of the function has a horizontal linear asymptote .

When and , the function will behave like ,

where .

The graph of the function goes to infinity or negative infinity.

If the degree of the numerator is one more than the degree of the denominator ( ), then the graph of the rational function will have an oblique asymptote.

Example 5 (Optional).

Find an equation of the linear oblique asymptote to the curve

.

Solution: Note the degree of the numerator is 1 more than the degree of the denominator.

Divide the numerator by the denominator.

Write the result as a division statement

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MHF 4U Unit 10: RATIONAL FUNCTIONS

If x approaches infinity or negative infinity, the value of

approaches zero. Then the curve approaches the line , which is a linear oblique asymptote.

The graph shows the curve and

the oblique asymptote.

Also we would expect to see a

vertical asymptote at x=2,

resulting from setting the

denominator equal to 0.

Example 6 Graph the function

Solution:

1. Represent the function in factored form:

2. domain f: .

3. The function change sign at x = -5, -3, 4 and 6. For x>6, f(x) is positive. This leads to the following diagram for the sign of f.

4. Vertical asymptotes at x= -3 and

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MHF 4U Unit 10: RATIONAL FUNCTIONS

5. x-intercepts: (-5, 0) and (4, 0); y-intercept: (0, )

6. Looking at the equation it is not clear whether the function has any

local maxima or minima.

7. As , the function

acts like and its graph

approaches the line y= 1.

Horizontal asymptote: y = 1.

Here is how the function looks on

the graph.

Curve Sketching

Here is a check-list of the things to consider when you make a sketch of the graph .

1. Factor the numerator and denominator (into linear factors if possible). Simplify.

2. Find the domain of .

3. Determine where the function is negative and where positive.

4. Find the points of discontinuity and their types – vertical asymptotes or donut holes.

5. Find the x- and y-intercepts.

6. Find the local maxima and local minima of the function. (Later you will learn how to do this. For now use your graphing calculator.)

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MHF 4U Unit 10: RATIONAL FUNCTIONS

7. Determine the end behaviour of the function and see whether the function has any horizontal or oblique asymptotes.

8. Examine the symmetry of the curve with respect to the y-axis and with respect to the origin, i.e. whether the function is even or odd.

pg. 272 # 1, 2 pg. 273 # 5 pg. 274 # 9, 14

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Assignment C: GRAPHS OF RATIONAL FUNCTIONS