Polynomials and Rational Functions

5-4 Rational Functions

Take notes!

A rational function is a function whose rule can be written as a ratio of two polynomials.  

These are some of the rational functions you can see in this chapter!

A discontinuous function is a function whose graph has one or more gaps or breaks. Many rational functions are discontinuous.

A continuous function is a function whose graph has no gaps or breaks. This means you can draw the entire thing without picking up your pencil. The functions we have looked at before this (linear, quadratic, polynomial, exponential, logarithmic) are all continuous.

Multiple Choice

The pictured graph is

• discontinuous

• continuous

Multiple Choice

The pictured graph is

• discontinuous

• continuous

Multiple Choice

The pictured graph is

• discontinuous

• continuous

The rational parent function is $f\left(x\right)=\frac{1}{x}$ .   

Its graph is a hyperbola, which has two separate branches.  

Reminder: an asymptote is an imaginary line that the graph will approach but never touch.

Like logarithmic and exponential functions, rational functions may have asymptotes.  

• The parent function has a vertical asymptote at x=0.

• The parent function has a horizontal asymptote at y=0.

The vertical asymptote at x=0 means the branches of this function will get really close but never actually touch the y-axis.

The domain of this function is all x's except for x=0, since there is an asymptote at x=0.

The graph is undefined at x=0! Remember that word??

The domain can be written multiple ways:

•  $\left(-\infty,0\right)\cup\left(0,\infty\right)$  (interval notation from left to right)

•  $\left\{x:\ x\ne0\right\}$  (all x's except 0)

The "union" in the middle of the interval notation just means there are multiple pieces of this domain.

The horizontal asymptote at y=0 means the branches of this function will get really close but never actually touch the x-axis.

The range of this function is all y's except for y=0, since there is an asymptote at y=0.

The range can be written multiple ways:

•  $\left(-\infty,0\right)\cup\left(0,\infty\right)$  (interval notation from bottom to top)

•  $\left\{y:\ y\ne0\right\}$  (all y's except 0)

When you transform the rational parent function, the asymptotes move too!

Multiple Choice

Select one of the transformations in

$$h\left(x\right)=\frac{1}{x-3}-5$$ .

• right 3

• down 3

• up 3

• left 3

Multiple Choice

Select one of the transformations in

$$h\left(x\right)=\frac{1}{x-3}-5$$ .

• right 5

• down 5

• up 5

• left 5

Multiple Choice

What x-value is not included in the domain of

$$h\left(x\right)=\frac{1}{x-3}-5$$ ? 

• 3

• 5

• 1

• -5

• -3

Multiple Choice

What y-value is not included in the range of

$$h\left(x\right)=\frac{1}{x-3}-5$$ ? 

• 3

• 5

• 1

• -5

• -3

Make sure you write this down.

Multiple Select

Select all true statements. You should have just taken notes on this on the last slide.

• Setting the numerator equal to zero gives us the vertical asymptotes.

• Setting the numerator equal to zero gives us the zeros.

• Setting the denominator equal to zero gives us the vertical asymptotes.

• Setting the denominator equal to zero gives us the zeros.

• Setting the denominator equal to zero gives us the horizontal asymptotes.

Examples

•  $f\left(x\right)=\frac{x^3}{x^2-4}$  has no horizontal asymptote

•  $g\left(x\right)=\frac{6x^2-6}{4x^3}$  has a horizontal asymptote at y=0

•  $j\left(x\right)=\frac{2x^2-7x-4}{3x^2+8x+5}$  has a horizontal asymptote at y=2/3

Multiple Choice

True or False:

If the degree of the numerator is smaller than the degree of the denominator, the horizontal asymptote is y=0.

You should have just taken notes on this.

• True

• False

Multiple Choice

True or False:

If the degree of the numerator is greater than the degree of the denominator, the horizontal asymptote is y= the leading coefficient of the numerator divided by the leading coefficient of the denominator.

• True

• False

Multiple Choice

True or False:

If the degree of the numerator is greater than the degree of the denominator, the horizontal asymptote is y=1.

• True

• False

Multiple Choice

True or False:

If the degree of the numerator is the same as the degree of the denominator, the horizontal asymptote is y= the leading coefficient of the numerator divided by the leading coefficient of the denominator.

• True

• False

If a rational function has the same factor in the numerator and denominator, they cancel and there is a hole at that x-value in the graph (instead of a vertical asymptote or zero).

Please please please make sure you have taken notes on everything in this presentation. I know it's a lot but having these notes will be very useful as you learn how to graph these functions!