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# Rational Functions Graphing Help

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By McGraw-Hill Professional
Updated on Oct 4, 2011

## Introduction to Rational Functions Graphing

When sketching the graph of a rational function, we use dashed lines for the asymptotes. We will sketch the graphs of rational functions in much the same way we sketched the graphs of polynomial functions. In addition to the points we plot for polynomial functions, we need to plot points to illustrate the asymptotic behavior of the graph. To show how a graph behaves near a vertical asymptote, we need to plot a point to its left and to its right. To show how a graph behaves near a horizontal asymptote, we need to plot points with large enough x -values, both positive and negative, to show how the graph flattens out. When a graph has both horizontal and vertical asymptotes, we will also plot a couple of mid-sized x -values.

#### Examples

Sketch the graph of the rational function.

The x -intercept is , the y -intercept is . The vertical asymptote is x = 4, and the horizontal asymptote is y = 2. We will use dashed lines for the asymptotes and plot the points for x = 3, x = 5, x = −10, and x = 10 to show how the graph behaves near the asymptotes.

Fig. 8.2

It is not obvious what the graph looks like so we will plot a point for x = 7. Then we will draw a smooth curve between the points.

Fig. 8.3

• There is no vertical asymptote because x 2 + 1 = 0 has no real solution. The x -axis is the horizontal asymptote. This graph has no x -intercept. The y -intercept is 1. We will use x = 5, − 5 to show the graph’s horizontal asymptotic behavior. The function is even, so the left half is a reflection of the right half. We will plot points for x = 1, 2. The y -values for x = − 1, −2 will be the same.

Fig. 8.4

• The vertical asymptotes are x = −1 and x = 1. The horizontal asymptote is y = 1. There is no x -intercept, and the y -intercept is −1. We will use x = 5, − 5 for the horizontal asymptote and x = −0.9, 0.9, −1.1, 1.1 for the vertical asymptotes. To get a better idea of what the graph looks like, we will need to plot other points. We will use x = 2 and x = −2.

Fig. 8.5

## Graphing Slant Asymptote

### Finding the Equation for Slant Asymptote

If the degree of the numerator is exactly one more than the degree of the denominator, then the graph has a slant asymptote. We can find the equation of a slant asymptote (a line whose slope is a nonzero number) by performing polynomial division. The equation for the slant asymptote is y = quotient.

#### Examples

Find an equation for the slant asymptote.

• When we divide 4 x 2 + 3 x − 5 by x + 2, we get a quotient of 4 x − 5. The slant asymptote is the line y = 4 x − 5.

•

The slant asymptote is y = x + 1.

### Graphing Slant Asymptote

When sketching the graph of a rational function that has a slant asymptote, we can show the behavior of the graph near the slant asymptote by plotting points for larger x -values. We can tell if an x -value is large enough by checking its y -values in both the line and rational function. If they are fairly close, then the x -value is large enough.

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