# Chapter 2 - Section 2.6 - Rational Functions and Their Graphs - Exercise Set - Page 399: 62

The graph of the function is:

#### Work Step by Step

Step 1: Substitute $x=-x$. \begin{align} & f\left( x \right)=\frac{4{{x}^{2}}}{{{x}^{2}}-9} \\ & f\left( -x \right)=\frac{4{{\left( -x \right)}^{2}}}{{{\left( -x \right)}^{2}}-9} \\ & =\frac{4{{x}^{2}}}{{{x}^{2}}-9} \\ & =f\left( x \right) \end{align} Therefore, the function $f\left( -x \right)$ is equal to $f\left( x \right)$. So, the graph of the function is either symmetrical about the $y$ axis or origin. Step 2: To calculate the x intercepts equate $f\left( x \right)=0$. \begin{align} & \frac{4{{x}^{2}}}{{{x}^{2}}-9}=0 \\ & x=0 \end{align} Step 3: To calculate the y intercepts evaluate $f\left( 0 \right)$ \begin{align} & f\left( 0 \right)=\frac{4\left( 0 \right)}{\left( 0 \right)-9} \\ & f\left( 0 \right)=0 \\ \end{align} Step 4: Since the degree of the numerator is equal to the denominator, the horizontal asymptote is: $y=4$. Step 5: For the vertical asymptote, equate the denominator to 0. \begin{align} & {{x}^{2}}-9=0 \\ & x=\pm 3 \end{align}

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