Points: 0 of 2 Save Follow the seven step strategy to graph the following rational function. \[ f(x)=\frac{4 x^{2}}{x^{2}-4} \] A. The \( x \)-intercept(s) is/are 0 . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There is no \( x \)-intercept. Find the vertical asymptote(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The equation(s) of the vertical asymptote(s) is/are \( x=2, x=-2 \). (Type an equation. Use a comma to separate answers as needed.) B. There is no vertical asymptote. Find the horizontal asymptote(s). Type the equation(s) of the horizontal asymptote(s). Select the correct choice below and fill in any answer boxes within your choice. A. The equation(s) of the horizontal asymptote(s) is/are \( \square \) . (Type an equation. Use a comma to separate answers as needed.) B. There is no horizontal asymptote.

To graph the rational function \( f(x) = \frac{4x^2}{x^2 - 4} \), we will follow the seven-step strategy. Let's break it down step by step. ### Step 1: Find the \( x \)-intercepts The \( x \)-intercepts occur where \( f(x) = 0 \). This happens when the numerator is zero. \[ 4x^2 = 0 \] Solving for \( x \): \[ x^2 = 0 \implies x = 0 \] Thus, the \( x \)-intercept is: **A. The \( x \)-intercept(s) is/are:** \( 0 \). ### Step 2: Find the vertical asymptotes Vertical asymptotes occur where the denominator is zero (and the numerator is not zero at those points). Set the denominator equal to zero: \[ x^2 - 4 = 0 \] Factoring gives: \[ (x - 2)(x + 2) = 0 \] Thus, the solutions are: \[ x = 2 \quad \text{and} \quad x = -2 \] So, the vertical asymptotes are: **A. The equation(s) of the vertical asymptote(s) is/are:** \( x = 2, x = -2 \). ### Step 3: Find the horizontal asymptotes To find the horizontal asymptote, we compare the degrees of the numerator and denominator. Both the numerator and denominator are degree 2. For rational functions where the degrees of the numerator and denominator are equal, the horizontal asymptote is given by the ratio of the leading coefficients. The leading coefficient of the numerator \( 4x^2 \) is \( 4 \), and the leading coefficient of the denominator \( x^2 - 4 \) is \( 1 \). Thus, the horizontal asymptote is: \[ y = \frac{4}{1} = 4 \] So, the horizontal asymptote is: **A. The equation(s) of the horizontal asymptote(s) is/are:** \( y = 4 \). ### Summary of Results - **\( x \)-intercept(s):** \( 0 \) - **Vertical asymptote(s):** \( x = 2, x = -2 \) - **Horizontal asymptote(s):** \( y = 4 \)