As you observed in the introduction, rational functions typically have asymptotes, and sometimes have holes.

Vertical asymptotes, horizontal asymptotes, and holes in the graph cause some of the variation in the graph. In the applet below, you will explore how varying the parameters in a rational function can affect the asymptotic behavior of the graph of the function.

In this section, you will investigate rational functions of the following form:

Rational functions of this form are ratios of two quadratic expressions, so the degree of the numerator and the degree of the denominator are both equal to 2. The quadratic expressions are presented in factored form to make it easier to identify patterns.

Click on the image to access the applet. Use the sliders to change the values of a, b, c, and d in the applet. Use the applet to complete the table below. Use the data in the table to answer the conclusion questions that follow.

Interactive popup. Assistance may be required.

Interactive popup. Assistance may be required.

### Conclusion Questions

• What relationship do you notice between the values of c and d and the equations of any vertical asymptotes?
• Interactive popup. Assistance may be required.

The vertical asymptotes are the lines x = c and/or x = d.
• What relationship do you notice between the values of a and b and the coordinates of the x-intercepts?
• Interactive popup. Assistance may be required.

The coordinates of the x-intercepts are (a, 0) and/or (b, 0).
• What had to be true for a hole to be present on the graph of the rational function?
• Interactive popup. Assistance may be required.

There had to be a common factor in the numerator and denominator. The value of x that made that common factor equal zero was the x-value of the coordinates of the hole in the graph.
• What caused the graph to split into three sections?
• Interactive popup. Assistance may be required.

When there were two vertical asymptotes, the graph was split into three sections.

### Pause and Reflect

As you worked through this section, the applet provided you with a chance to see several types of discontinuity in the graph of a rational function. What are two types of discontinuity in which the graph is undefined?

Interactive popup. Assistance may be required.

The vertical asymptotes and holes in the graph are two types of discontinuity in which the graph is undefined.

Finding x-intercepts is something that you have done in previous lessons. Without looking at the graph of a rational function, how can you find the x-intercepts?

Interactive popup. Assistance may be required.

Setting the numerator equal to zero and solving for x will give you the x-intercepts of the rational function. However, common factors in the numerator and denominator can create holes in the graph. These holes will cause an x-intercept to be undefined since the denominator cannot equal zero at that point.

### Practice

1. Jerome graphed a rational function that has a hole at x = 2. What common factor appears in both the numerator and denominator of Jerome's rational function?
Interactive popup. Assistance may be required.

Think back to the table and the applet. What did each function that had a hole have in common?
Interactive popup. Assistance may be required.

x – 2
2. Julianne is looking at the rational function, g(x) = x(x + 3) over (x - 2)(x - 1) x(x + 3) (x – 2)(x – 1) . Before graphing the function, what can Julianne predict about the asymptotes, holes, and x-intercepts of this function?

Interactive popup. Assistance may be required.

From the applet in this section, think about what would happen if a = 0, b = -3, c = 2 and d =1.
Interactive popup. Assistance may be required.

Julianne could say that there would be x-intercepts at (0, 0) and (-3, 0) and vertical asymptotes at x = 2 and x = 1. There is also a horizontal asymptote at y = 1.
3. Simon knows that a rational function has x-intercepts at (-2, 0) and (3, 0) as well as vertical asymptotes at x = 1 and x = -4. What is a possible equation for this graph?

Interactive popup. Assistance may be required.

Recall that the x-intercepts can be found from the numerator of the function and its factors, while the vertical asymptotes can be found from the denominator and its factors.
Interactive popup. Assistance may be required.

f(x) = (x + 2)(x - 3) over (x - 1)(x + 4) (x + 2)(x – 3) (x – 1)(x + 4)