Question

follow the seven step strategy to graph the following rational function.
$f(x) = \frac{3x}{x^2 - 4}$

$f(x)=\frac{3x}{x^2 - 4}$ $-\frac{7}{15}$ $-\frac{9}{16}$ 1 -1 $\frac{5}{7}$ $\frac{9}{16}$
(simplify your answers.)

use the information obtained in the previous steps to graph the function between and beyond the vertical asymptotes. choose the correct graph below.
\\(\bigcirc\\) a. \\(\bigcirc\\) b. \\(\bigcirc\\) c. \\(\bigcirc\\) d.
(graphs of functions a, b, c, d are shown with coordinate grids and function curves.)

Explanation:

Step1: Analyze the function

The function is \( f(x)=\frac{3x}{x^2 - 4}=\frac{3x}{(x - 2)(x + 2)} \). Vertical asymptotes at \( x = 2 \) and \( x=-2 \), horizontal asymptote \( y = 0 \) (since degree of numerator < degree of denominator), and the y - intercept at \( x = 0 \), \( f(0)=0 \), x - intercept at \( x = 0 \). The function is odd ( \( f(-x)=\frac{-3x}{x^2 - 4}=-f(x) \) ), so symmetric about the origin.

Step2: Analyze the options

• Option A: Does not show symmetry about origin and the shape is wrong.
• Option B: The graph is a parabola - like shape, but our function is a rational function with vertical asymptotes, so B is wrong.
• Option C: The right - hand side of \( x = 2 \) and left - hand side of \( x=-2 \) do not match the behavior of an odd function.
• Option D: The graph is symmetric about the origin, has vertical asymptotes at \( x = 2 \) and \( x=-2 \), horizontal asymptote \( y = 0 \), and the behavior of the branches (as \( x

ightarrow2^{+} \), \( f(x)
ightarrow-\infty \); as \( x
ightarrow2^{-} \), \( f(x)
ightarrow+\infty \); as \( x
ightarrow - 2^{+} \), \( f(x)
ightarrow-\infty \); as \( x
ightarrow - 2^{-} \), \( f(x)
ightarrow+\infty \)) matches the odd - function property and the rational function's behavior.

Answer:

D. The graph of \( f(x)=\frac{3x}{x^2 - 4} \) (the fourth graph)