Final answer:
The equation for the final quadratic function is y = -0.5(x + 0.5)^2 + 10.125.
Step-by-step explanation:
The given quadratic function has x-intercepts at 4 and -5, and a vertex at (-0.5, 10.125). To find the equation for the quadratic function, we can start by using the vertex form of a quadratic equation: y = a(x - h)^2 + k, where (h, k) is the vertex. Plugging in the vertex coordinates, we have y = a(x - (-0.5))^2 + 10.125. To find the value of a, we can substitute one of the x-intercepts into the equation. Let's use the x-intercept 4. Plugging in x = 4 and y = 0 into the equation, we get 0 = a(4 - (-0.5))^2 + 10.125. Solving this equation will give us the value of a.
First, let's simplify the equation: 0 = a(4 + 0.5)^2 + 10.125. Squaring the terms inside the parentheses: 0 = a(4.5)^2 + 10.125. Simplifying further: 0 = a(20.25) + 10.125. Distributing the a: 0 = 20.25a + 10.125. Moving the constant term to the other side: -10.125 = 20.25a. Dividing both sides by 20.25, we can find the value of a: a = -10.125/20.25. Simplifying this expression, we get a = -0.5.
Now that we have the value of a, we can substitute it back into the equation y = a(x - h)^2 + k: y = -0.5(x - (-0.5))^2 + 10.125. Simplifying the equation further, we have y = -0.5(x + 0.5)^2 + 10.125. This is the equation for the final quadratic function, so the correct answer is a) y = -0.5(x + 0.5)^2 + 10.125.