CPMSection 7.1

“The Rational Function”

In Chapter 4, we discussed the linear function. In Ch. 5, it was the absolute value function and in Chapter 6 the exponential function.

Each of these functions have a behavior called continuity. A function is

continuous if its graph ____________________________________________. As

you will see, rational functions do not have this behavior.

A rational function is a function in which ___________________________________. The general form of a rational

function is y=_______________.

can be traced, in its entirety, w/out lifting your pencil

one polynomial is divided by a secondd

fx e

We will examine how each of the parameters d, e, and f affect the graph later in these notes. For now, lets take an example. The simplest rational function, the parent function is . Graph it.1

yx

X -4 -3 -2 -1 0 1 2 3 4

Y -.25 -.33 -.5 -1 E 1 .5 .33 .25

asymptotes

A rational function is undefined at all values of the variable x that ___________________________. The graph of the function will have a ____________________ at these values of x. The rational function will be undefined when x = _____.The domain of a rational function is _________________________________________(i.e._____).The parameter f determines _____________________________.

The range of the function will be ________________.

If the parameter d is negative, then _____________________________________.

makes the denominator = 0vertical asymptote

-e

All real #’s, except those that make the function undefined -e

where the horizontal asymptote occurs

the graph flips over the horizontal asymptote

, y f

Let’s look at the function

What is our Vertical Asymptote?

What is our Horizontal Asymptote?

Let’s find the graph on our calculator

13

5y

x

x = 5

y = 3

Asymptotes

What is the RANGE of our graph?

All Real #’s, y≠3What is the DOMAIN of our graph?

All Real’s, x≠5Where is our Y-Intercept?

Set x=0, solve for y Y-Int = 2.8

What values of x make y = 0?(This is the zero function)

x = 4.67

For what values of x is our function increasing?

NoneFor what values of x is our function decreasing?

(-∞, 5) (5, ∞)As x goes to 5 from the right, y goes to what number?

InfinityAs x goes to 5 from the left, y goes to what number?

Negative Infinity