Write an equation for a rational function with the following properties: - Vertical asymptotes at \( x=1 \) and \( x=-4 \) - Zeros ( \( x \)-intercepts) at \( x=-2 \) and \( x=-5 \) - A \( y \)-intercept at \( (0,5) \) Leave your equation in foctored form. \( r(x)=\square \)

To construct a rational function \( r(x) \) based on the given properties, we can start by writing the factors for the vertical asymptotes and the zeros. The vertical asymptotes at \( x=1 \) and \( x=-4 \) indicate the function will have factors in the denominator: \[ (x - 1)(x + 4) \] The zeros at \( x=-2 \) and \( x=-5 \) indicate that the function will have factors in the numerator: \[ (x + 2)(x + 5) \] Now, putting these factors together, we have: \[ r(x) = \frac{(x + 2)(x + 5)}{(x - 1)(x + 4)} \] Next, we need to ensure that the \( y \)-intercept is \( (0, 5) \). To find the \( y \)-intercept, we can substitute \( x = 0 \) into our function: \[ r(0) = \frac{(0 + 2)(0 + 5)}{(0 - 1)(0 + 4)} = \frac{(2)(5)}{(-1)(4)} = \frac{10}{-4} = -\frac{5}{2} \] This gives us \( -\frac{5}{2} \) at \( x=0 \). To adjust our function to achieve a \( y \)-intercept of 5, we need to introduce a constant \( k \) in the numerator: \[ r(x) = k \cdot \frac{(x + 2)(x + 5)}{(x - 1)(x + 4)} \] We want \( r(0) = 5 \): \[ 5 = k \cdot \left(-\frac{5}{2}\right) \Rightarrow k = -2 \] So, we multiply through by -2: \[ r(x) = -2 \cdot \frac{(x + 2)(x + 5)}{(x - 1)(x + 4)} \] Thus, the final rational function in factored form is: \[ r(x) = \frac{-2(x + 2)(x + 5)}{(x - 1)(x + 4)} \]