Write an equation for a parabola with \( x \)-intercepts \( (-3,0) \) and \( (2,0) \) which passes through the point \( (1,-8) \). Write the equation.

To write the equation of a parabola with given \( x \)-intercepts, we can start with the factored form of the quadratic equation. Since the \( x \)-intercepts are \( (-3, 0) \) and \( (2, 0) \), we can express the equation as: \[ y = a(x + 3)(x - 2) \] where \( a \) is a constant that will determine the vertical stretch or compression of the parabola. Next, we need to find the value of \( a \) using the point \( (1, -8) \), which lies on the parabola. We can substitute \( x = 1 \) and \( y = -8 \) into the equation: \[ -8 = a(1 + 3)(1 - 2) \] Calculating the terms inside the parentheses: \[ -8 = a(4)(-1) \] This simplifies to: \[ -8 = -4a \] Now, we can solve for \( a \): \[ a = \frac{-8}{-4} = 2 \] Now that we have the value of \( a \), we can write the complete equation of the parabola: \[ y = 2(x + 3)(x - 2) \] To express this in standard form, we can expand it: \[ y = 2[(x + 3)(x - 2)] = 2[x^2 - 2x + 3x - 6] = 2[x^2 + x - 6] \] Now, distributing the \( 2 \): \[ y = 2x^2 + 2x - 12 \] Thus, the equation of the parabola is: \[ \boxed{y = 2x^2 + 2x - 12} \]