rational functions and their graphs

Rational Functions and Their Graphs

Objectives

Find the domain of rational functions. Find horizontal and vertical asymptotes

of graphs of rational functions. Analyze and sketch graphs of rational

functions.

Rational Functions

• A rational function is the ratio of two polynomial functions

• Asymptote comes from combining 3 Greek words “an-sum-piptein” meaning “does not fall together with”. An asymptote is a curve that another curve approaches but does not ultimately cross.

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)()(

xq

xpxf

Finding the domain of a rational function

Rational functions contain division. Because division by zero is undefined, we must solve for the polynomial in the denominator for if it equals zero.

Solve: x − 3 = 0 x = 3 (the restriction)

Domain of f = {x│x ≠ 3) set-builder notation

Domain of f = (− ∞, 3) ⋃ (3, ∞) interval notation

The Domain of is all real numbers except 3.

Find the domain of the rational function

Answer in set-builder notation is

Answer in interval notation is

Solve for 𝑥=3∧𝑥=− 3

{𝑥∨𝑥 ≠3∧− 3 }

(− ∞ ,− 3)∪ (−3 ,3)∪ (3 , ∞ )

The domain of is all real numbers except

Find the domain of the rational function

Answer in set-builder notation is

Answer in interval notation is

Since we don’t have a restriction, the domain will be all real numbers.

{𝑥∨𝑥=all real numbers }

(− ∞ , ∞)

The Reciprocal Function

The domain of this function is all real numbers except zero.

Let’s look at the behavior of this function on the left of zero.


x is approaching zero from the left. As x approaches 0 from the left, what are the y-values, f(x), doing? They are decreasing without bound towards negative infinity.

− ∞ ∞

∞

− ∞


We use a special arrow notation to describe this situation.

As f(x) → − ∞

This is read, “As x approaches 0 from the left, f(x) approaches negative infinity.”

− ∞ ∞

∞

− ∞


x is approaching zero from the right. As x approaches 0 from the right, what are the y-values, f(x), doing? They are increasing without bound towards positive infinity.

− ∞ ∞

∞

− ∞


We use a special arrow notation to describe this situation.

As , f(x) → ∞

This is read, “As x approaches 0 from the right, f(x) approaches infinity.”

− ∞ ∞

∞

− ∞

Arrow Notation

Symbol Meaning

x approaches from the right

x approaches from the left

x approaches infinity

x approaches negative infinity

Vertical Asymptotes

Refer to page 343 in your book. The line x = a is a vertical asymptote of the graph of a function f if f(x) increases or decreases without bound as x approaches a.

A graph may have more than one vertical asymptote. It may have no vertical asymptotes.

Look at the four examples on page 343. The dotted lines represent the asymptotes. Think of them as an electric fence which can be approached closely but cannot be touched.

Locating Vertical Asymptotes

Step 1: Simplify the rational function, if possible.

Step 2: Solve for the polynomial in the denominator for if it equals zero.

Find the vertical asymptotes, if any, of the graph of:

Check out the graph on the graphing calculator.

Locating Vertical Asymptotes

Find the vertical asymptotes, if any, of the graph of:

Check out graph on the graphing calculator.

Step 1: Simplify the rational function, if possible.

Step 2: Solve for the polynomial in the denominator for if it equals zero.

Graphing a Rational Function with a Hole

Example: Graph

Solution: Notice the domain of the function cannot include 2.

Rewrite f in lowest terms by factoring the numerator.

.24

)(2

xx

xf

)2(22

)2)(2(24

)(2

xxx

xxxx

xf

The graph of f is the graph of the line y = x + 2 with the exception of the point with x-value 2.

The line y = b is a horizontal asymptote of the graph of a function f if f(x) approaches b as x increases or decreases without bound.

Horizontal Asymptotes

The dotted lines represent the asymptote. There can at most only be one horizontal asymptote. The graph may cross its horizontal asymptote.

As As As

Locating Horizontal Asymptotes

The degree of the numerator is n. The degree of the denominator is d.

If n < d, the x-axis, or y = 0, is the horizontal asymptote of the graph. If n = d, the horizontal asymptote will be:

If n > d, the graph has no horizontal asymptotes.

Find the horizontal asymptote:

Compare the lead degrees of the two polynomials.

The degree of the numerator is less than the degree of the denominator.

If n < d, the x-axis, or y = 0, is the horizontal asymptote of the graph.

Find the horizontal asymptote:


The degree of the numerator is the same as the degree of the denominator.

If n = d, the horizontal asymptote will be:

; The horizontal asymptote is .

Find the horizontal asymptote:hCompare the lead degrees of the two polynomials.

The degree of the numerator is greater than the degree of the denominator.

If n > d, the graph has no horizontal asymptotes.

Graphing Rational Functions

1. Simplify, if possible, the rational function.2. Find the y-intercept (if there is one) by substituting 0

for every x-variable.3. Find the x-intercepts (if there are any) by letting f(x) be

zero.4. Find any vertical asymptotes.5. Find the horizontal asymptotes (if there is one).6. Plot several points between and beyond each x-

intercept and vertical asymptote.7. Complete the sketch.

Graphing A Rational Function

Step 1: Simplify, if possible, the rational function. Is it simplified? Yes

Step 2: Find the y-intercept (if there is one) by substituting 0 for every x-variable. What is the y-intercept?1

Step 3: Find the x-intercepts (if there are any) by letting f(x) be zero. The x-intercept is 1

2


Step 4: Find any vertical asymptotes.

𝑥=1Step 5: Find the horizontal asymptotes (if there is one).

Solve for the polynomial in the denominator for if it equals zero.


If n = d, the horizontal asymptote will be 𝑦=2


Step 6: Plot several points between and beyond each x-intercept and vertical asymptote.

Final Step: Complete the sketch.

rational functions and their graphs

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