Question: Write an equation for a rational function with: Vertical asymptotes at \( x = 1 \) and \( x =…

Write an equation for a rational function with:

Vertical asymptotes at \( x = 1 \) and \( x = -5 \)

\( x \) intercepts at \( x = -1 \) and \( x = -6 \)

Horizontal asymptote at \( y = 3 \)

\( y = \boxed{\quad} \)

Solution

Identify the vertical asymptotes and write the denominator. \[ (x - 1)(x + 5) & \] Identify the \(x\)-intercepts and write the numerator. \[ (x + 1)(x + 6) & \] Determine the leading coefficients based on the horizontal asymptote \(y = 3\). \[ \frac{3(x + 1)(x + 6)}{(x - 1)(x + 5)} & \] Write the equation of the rational function. \[ y = \frac{3(x + 1)(x + 6)}{(x - 1)(x + 5)} \]