Questions: Use your knowledge of asymptotes and intercepts to match the equation, f(x) = 3x / (x^2 - 25), with one of the graphs that follow. List all the asymptotes. Check your work using a graphing calculator.

Solution Steps

To find the vertical asymptotes of the function \( f(x) = \frac{3x}{x^2 - 25} \), we set the denominator equal to zero and solve for \( x \):

\[ x^2 - 25 = 0 \]

Solving for \( x \):

\[ x^2 = 25 \implies x = \pm 5 \]

Thus, the vertical asymptotes are at \( x = -5 \) and \( x = 5 \).

To determine the horizontal asymptote, we compare the degrees of the numerator and the denominator. The degree of the numerator \( 3x \) is 1, and the degree of the denominator \( x^2 - 25 \) is 2. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is:

\[ y = 0 \]

To verify the asymptotes, we can plot the function \( f(x) = \frac{3x}{x^2 - 25} \) and observe the behavior near the asymptotes. The graph confirms that the function approaches the vertical asymptotes at \( x = -5 \) and \( x = 5 \), and the horizontal asymptote at \( y = 0 \).

Final Answer