A rational function is a ratio of two polynomials.

For example, the function has a polynomial of degree 0 in its numerator, and a polynomial of degree 1 in its denominator. So, f(x) is a rational function.

Notice that this equation is not defined when x = 0, because the denominator would be equal to zero. So the domain of this function is all real numbers except zero.

You can see that the graph is defined for all x-values except zero. So this verifies that the domain of the function is the set of all real numbers except zero.

Click the checkbox under the slider for q to add the factor (x - q). Then use the sliders to change the values of p and q in the equation .

The graph includes all y-values except zero. So the range of the function is all real numbers except zero.

Let's explore the graphs of other rational functions.

The graph of is displayed to the right. x = 2 is the vertical asymptote of this graph. This is because f(x) is undefined when its denominator is 0.

Use the tool to change the value of p. Notice how changing p affects the graph of f(x).

You can choose to add or remove either the (x - p) or the (x - q) factor from the equation by clicking the checkbox.

Think about the following questions as you explore: How do the values of p and q affect the graph? How many vertical asymptotes does the graph have? What is the horizontal asymptote? Click "Done Exploring" when finished.

A function of the form has vertical asymptotes at x = p and x = q, and a horizontal asymptote at y = 0.

Now let's see what happens when we add a factor to the numerator of f(x).

Click the box under the slider for r to add the factor (x - r) to the numerator. Then use the sliders to change the values of p, q, and r.

Think about the following questions: Does adding a factor to the numerator affect the vertical asymptotes? What about the horizontal asymptote? Does it affect the x-intercepts? When does a hole appear?

Again, you can choose to add or remove any of the factors from the equation by clicking their checkboxes. How does the graph change when there is only one factor in the denominator instead of two? Click "Done Exploring" when finished.

For values that are undefined but are not asymptotes, a hole will appear on the graph.

The graph of a rational function has a hole at a value of x that makes both the numerator and denominator equal to zero.

The graph crosses the x-axis at a value of x that makes only the numerator equal to zero.

The domain of a rational function consists of all real values of x except those that make the denominator zero. The graph of the function has vertical asymptotes or holes at these values.

Which of the functions to the right crosses the x-axis and has two vertical asymptotes? Click "Submit" when finished.

You have seen how the factors of a rational function affect its graph. Now let's look at the effect of multiplying a rational function by a constant factor.

Use the slider to change the value of k in the equation . Notice how changing this constant factor affects the graph.

Think about the following questions as you explore: Does changing k affect the location of the x-intercept? Does it affect the y-intercept? Does it affect the vertical asymptote? What about the horizontal asymptote? Click "Done Exploring" when finished.

Multiplying this equation by a constant causes its graph to be stretched or compressed vertically. The x-intercept and vertical asymptote do not change. However, the y-intercept and the horizontal asymptote do change.

When x takes on very large positive or negative values, the value of f(x) approaches k. So the horizontal asymptote for this function is the line y = k.

Multiplying a rational function by a constant stretches or compresses its graph vertically. This can affect the horizontal asymptote of the function.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education