Questions: Find the slant asymptote of the graph of the rational function. Follow the seven-step strategy and use the slant asymptote to graph the rational function. f(x) = (x^2 - 25) / x A. The equation(s) of the horizontal asymptote(s) is/are □ (Type an equation. Use a comma to separate answers as needed.) B. There is no horizontal asymptote Plot points between and beyond the x-intercepts and the vertical asymptote. Evaluate the function at -7, -4, 4, 7, and 8. x -7 -4 4 7 8 f(x) = (x^2 - 25) / x □ □ □ □ □ (Simplify your answers.)

Solution Steps

The slant asymptote of a rational function \( f(x) = \frac{p(x)}{q(x)} \) occurs when the degree of the numerator is exactly one more than the degree of the denominator. Here, \( f(x) = \frac{x^2 - 25}{x} \).

Perform polynomial long division: \[ x^2 - 25 \div x = x - 0 + \frac{-25}{x} \] The slant asymptote is \( y = x \).

Evaluate \( f(x) = \frac{x^2 - 25}{x} \) at \( x = -7, -4, 4, 7, 8 \).

• \( f(-7) = \frac{(-7)^2 - 25}{-7} = \frac{49 - 25}{-7} = \frac{24}{-7} = -\frac{24}{7} \)
• \( f(-4) = \frac{(-4)^2 - 25}{-4} = \frac{16 - 25}{-4} = \frac{-9}{-4} = \frac{9}{4} \)
• \( f(4) = \frac{4^2 - 25}{4} = \frac{16 - 25}{4} = \frac{-9}{4} \)
• \( f(7) = \frac{7^2 - 25}{7} = \frac{49 - 25}{7} = \frac{24}{7} \)
• \( f(8) = \frac{8^2 - 25}{8} = \frac{64 - 25}{8} = \frac{39}{8} \)

Final Answer