## Precalculus (6th Edition) Blitzer

Step 1: Substitute $x=-x$ \begin{align} & f\left( x \right)=\frac{2{{x}^{2}}}{{{x}^{2}}+4} \\ & f\left( -x \right)=\frac{2{{\left( -x \right)}^{2}}}{{{\left( -x \right)}^{2}}+4} \\ & =\frac{2{{x}^{2}}}{{{x}^{2}}+4} \\ & =f\left( x \right) \end{align} Therefore, the function $f\left( -x \right)$ is equal to $f\left( x \right)$. Hence, the graph of the function is symmetric about the $y$ -axis. Step 2: To calculate the x intercepts equate $f\left( x \right)=0$. \begin{align} & \frac{2{{x}^{2}}}{{{x}^{2}}+4}=0 \\ & x=0 \end{align} , Step 3: To calculate the y intercept evaluate $f\left( 0 \right)$. $f\left( 0 \right)=0$ Step 4: Since the degree of the numerator is equal to the denominator, the horizontal asymptote is: $y=2$ Step 5: For the vertical asymptote, equate the denominator to 0. The function has no real asymptotes.