Final answer:
The equation of the parabola with x-intercepts at (2,0) and (3,0), and y-intercept at (0,-12) is y = -2x^2 + 10x - 12.
Step-by-step explanation:
To find the equation of a parabola with x-intercepts at (2,0) and (3,0), and a y-intercept at (0,-12), we can use the factored form of a quadratic equation, which is y = a(x - h)(x - k), where (h,0) and (k,0) are the x-intercepts.
Substituting our known intercepts, we get y = a(x - 2)(x - 3). To find a, we use the y-intercept (0,-12), giving us -12 = a(0 - 2)(0 - 3), which simplifies to -12 = 6a. Solving for a gives us a = -2.
Thus, the final equation of the parabola is y = -2(x - 2)(x - 3), which can be expanded to y = -2(x^2 - 5x + 6), and then y = -2x^2 + 10x - 12 after distributing the -2.