5.6 Rational functions  (Page 7/16)

Finding the intercepts of a rational function

Find the intercepts of $f(x)=\frac{(x-2)(x+3)}{(x-1)(x+2)(x-5)}.$

We can find the y -intercept by evaluating the function at zero

$$\begin{array}{ccc}\hfill f(0)& =& \frac{(0-2)(0+3)}{(0-1)(0+2)(0-5)}\hfill \\ & =& \frac{-6}{10}\hfill \\ & =& -\frac{3}{5}\hfill \\ & =& \mathrm{-0.6}\hfill \end{array}$$

The x -intercepts will occur when the function is equal to zero:

$$\begin{array}{cccc}\hfill 0& =& \frac{(x-2)(x+3)}{(x-1)(x+2)(x-5)}\hfill & \text{This is zero when the numerator is zero}.\hfill \\ \hfill 0& =& (x-2)(x+3)\hfill & \\ \hfill x& =& 2,\mathrm{-3}\hfill & \end{array}$$

The y -intercept is $(0,\mathrm{–0.6}),$ the x -intercepts are $(2,0)$ and $(\mathrm{–3},0).$ See [link] .

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Graphing rational functions

In [link] , we see that the numerator of a rational function reveals the x -intercepts of the graph, whereas the denominator reveals the vertical asymptotes of the graph. As with polynomials, factors of the numerator may have integer powers greater than one. Fortunately, the effect on the shape of the graph at those intercepts is the same as we saw with polynomials.

The vertical asymptotes associated with the factors of the denominator will mirror one of the two toolkit reciprocal functions. When the degree of the factor in the denominator is odd, the distinguishing characteristic is that on one side of the vertical asymptote the graph heads towards positive infinity, and on the other side the graph heads towards negative infinity. See [link] .

When the degree of the factor in the denominator is even, the distinguishing characteristic is that the graph either heads toward positive infinity on both sides of the vertical asymptote or heads toward negative infinity on both sides. See [link] .

For example, the graph of $f(x)=\frac{{(x+1)}^2(x-3)}{{(x+3)}^2(x-2)}$ is shown in [link] .

• At the x -intercept $x=\mathrm{-1}$ corresponding to the ${(x+1)}^2$ factor of the numerator, the graph "bounces", consistent with the quadratic nature of the factor.
• At the x -intercept $x=3$ corresponding to the $(x-3)$ factor of the numerator, the graph passes through the axis as we would expect from a linear factor.
• At the vertical asymptote $x=\mathrm{-3}$ corresponding to the ${(x+3)}^2$ factor of the denominator, the graph heads towards positive infinity on both sides of the asymptote, consistent with the behavior of the function $f(x)=\frac{1}{x^2}.$
• At the vertical asymptote $x=2,$ corresponding to the $(x-2)$ factor of the denominator, the graph heads towards positive infinity on the left side of the asymptote and towards negative infinity on the right side, consistent with the behavior of the function $f(x)=\frac{1}{x}.$