Question

the graph of ( f ) is shown below.
write an equation of the rational function that represents the graph.
( r(x) = )
submit question

Explanation:

Step1: Identify horizontal asymptote

The horizontal dashed line is $y=3$, so the rational function has a horizontal asymptote $y=3$, meaning the ratio of leading coefficients is 3. We start with the form $R(x)=3+\frac{a}{x-h}$, where $x=h$ is the vertical asymptote.

Step2: Identify vertical asymptote

The graph has a vertical asymptote at $x=0$ (the y-axis, since the graph approaches $\pm\infty$ as $x$ approaches 0), so $h=0$. Now the function is $R(x)=3+\frac{a}{x}$.

Step3: Solve for $a$ using a point

Use the point $(1, -7)$ (from the lower curve, when $x=1$, $y=-7$):
$$-7=3+\frac{a}{1}$$
$$a=-7-3=-10$$

Answer:

$R(x)=3-\frac{10}{x}$