CPM Section 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 second

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.

X Y E 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 = 0 vertical 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

Let’s look at the function What is our Vertical Asymptote? What is our Horizontal Asymptote? Let’s find the graph on our calculator x = 5 y = 3

Asymptotes

What is the RANGE of our graph? All Real #’s, y≠3 What is the DOMAIN of our graph? All Real’s, x≠5 Where 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? None For what values of x is our function decreasing? (-∞, 5) (5, ∞) As x goes to 5 from the right, y goes to what number? Infinity As x goes to 5 from the left, y goes to what number? Negative Infinity