Example 10 from Algebra and Trigonometry, 5.6 Rational Functions

Finding the Intercepts of a Rational Function

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

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

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

$\begin{matrix} f (0) & = & \frac{(0 - 2) (0 + 3)}{(0 - 1) (0 + 2) (0 - 5)} \\ = & \frac{- 6}{10} \\ = & - \frac{3}{5} \\ = & −0.6 \end{matrix}$
The x-intercepts will occur when the function is equal to zero:

$\begin{matrix} 0 & = & \frac{(x - 2) (x + 3)}{(x - 1) (x + 2) (x - 5)} & This is zero when the numerator is zero . \\ 0 & = & (x - 2) (x + 3) \\ x & = & 2, −3 \end{matrix}$

The y-intercept is $(0, –0.6),$ the x-intercepts are $(2, 0)$ and $(–3, 0) .$ See Figure 16.

Figure 16

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Given the reciprocal squared function that is shifted right 3 units and down 4 units, write this as a rational function. Then, find the x- and y-intercepts and the horizontal and vertical asymptotes.

How did it go?

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Given the reciprocal squared function that is shifted right 3 units and down 4 units, write this as a rational function. Then, find the x- and y-intercepts and the horizontal and vertical asymptotes.

Solution

Your turn

Solution

Your turn