Question: 6 — Graph of Rational Functions
LEARNING OBJECTIVE: Determine a vertical, horizontal, or oblique asymptote
of a rational function.
The graph of the rational function f(x) = (x - 2) / (x^2 + 4x - 5) will have a vertical
asymptote at __________.
a.) x = 1 and x = 2
b.) x = -5, x = 1, and x = 2
c.) x = 1 and x = -5
d.) x = -5 and x = 2
1. **State the problem:** We need to find the vertical asymptotes of the rational function $$f(x) = \frac{x - 2}{x^2 + 4x - 5}$$.
2. **Recall the rule for vertical asymptotes:** Vertical asymptotes occur where the denominator is zero and the numerator is not zero at those points.
3. **Find the zeros of the denominator:** Solve $$x^2 + 4x - 5 = 0$$.
4. **Factor the quadratic:** $$x^2 + 4x - 5 = (x + 5)(x - 1)$$.
5. **Set each factor equal to zero:**
$$x + 5 = 0 \implies x = -5$$
$$x - 1 = 0 \implies x = 1$$
6. **Check the numerator at these points:**
At $$x = -5$$, numerator $$= -5 - 2 = -7 \neq 0$$
At $$x = 1$$, numerator $$= 1 - 2 = -1 \neq 0$$
7. **Conclusion:** Both $$x = -5$$ and $$x = 1$$ make the denominator zero but not the numerator, so vertical asymptotes are at $$x = -5$$ and $$x = 1$$.
8. **Answer choice:** c.) $$x = 1$$ and $$x = -5$$