Question: Write an equation for a rational function with: Vertical asymptotes at \( x = 3 \) and \( x =…

Write an equation for a rational function with:

Vertical asymptotes at \( x = 3 \) and \( x = -3 \)

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

\( y \) intercept at 4

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

Solution

Start by identifying the factors based on the given characteristics. \[ y = \frac{k(x + 6)(x - 1)}{(x - 3)(x + 3)} \] Next, use the y-intercept to find the value of \( k \). When \( x = 0 \), \( y = 4 \): \[ 4 = \frac{k(0 + 6)(0 - 1)}{(0 - 3)(0 + 3)} \] Simplify the equation: \[ 4 = \frac{k \cdot 6 \cdot (-1)}{(-3) \cdot 3} \] \[ 4 = \frac{-6k}{-9} \] \[ 4 = \frac{6k}{9} \] \[ 4 = \frac{2k}{3} \] Solve for \( k \): \[ k = 6 \] Substitute \( k \) back into the original equation: \[ y = \frac{6(x + 6)(x - 1)}{(x - 3)(x + 3)} \]