Exercise 25 Page 526

Let's recall how to write a rational function given the following characteristics.

Graph Properties
$y -$ intercept Found when substituing $x = 0$ into the rational expression.		$x -$ intercept Occurs where the numerator equals zero.
Vertical Asymptote $x = a$ is a vertical asymptote if it is a zero for the denominator which has the factor $(x - a)$ and has no matching factor in the numerator.		Hole $x = a$ is a hole if $(x - a)$ is a common factor in the numerator and denominator and $a$ represents a zero of the rational expression.
Horizontal Asymptote with numerator degree $m$ and denominator degree $n$
If $m < n,$ the graph has a horizontal asymptote $y = 0 .$	If $m > n,$ the graph has no horizontal asymptote.	If $m = n,$ the graph has a horizontal asymptote at $y = \frac{a}{b},$ the ratio of the leading coefficients in the numerator and denominator.

Let's apply this knowledge towards our given characteristics. We will write a function applying the first characteristic, and then modify it so the second characteristic applies as well.

Given Characteristic	Interpretation	Possible Function
Hole at $x = - 5$	$(x + 5)$ in numerator and denominator	$\frac{x + 5}{x + 5}$
Vertical asymptote at $x = 2$	$(x - 2)$ in denominator only	$\frac{x + 5}{( x + 5 ) ( x - 2 )}$

We can see that the rational function

\frac{x + 5}{( x + 5 ) ( x - 2 )}

is an example of a function that satisfies the given characteristics. We can multiply our factors in the denominator to simplify our expression.

\frac{x + 5}{( x + 5 ) ( x - 2 )}

Distr

Distribute $(x + 5)$

\frac{x + 5}{x ( x + 5 ) - 2 ( x + 5 )}

Distr

Distribute $x$

\frac{x + 5}{x ^{2} + 5 x - 2 ( x + 5 )}

Distr

Distribute $- 2$

\frac{x + 5}{x ^{2} + 5 x - 2 x - 1 0}

AddSubTerms

Add and subtract terms

\frac{x + 5}{x ^{2} + 3 x - 1 0}