`y=a(x-2)(x+4)`
In the quadratic equation above, a is a nonzero constant. The graph of the equation in the xy-plane is a parabola with vertex (c,d) . Which of the following is equal to d ?

Options

A. `-9a`
B. `-8a`
C. `-5a`
D. `-2a`

Answer

To find the value of \( d \) in the quadratic equation \( y = a(x-2)(x+4) \), we will follow these steps:

### Step 1: Identify the Roots
The roots of the quadratic equation can be found by setting \( y = 0 \):
\[ a(x-2)(x+4) = 0 \]
This gives us the roots:
1. \( x - 2 = 0 \) → \( x = 2 \)
2. \( x + 4 = 0 \) → \( x = -4 \)

### Step 2: Find the x-coordinate of the Vertex
The x-coordinate of the vertex of a parabola can be found by taking the average of the roots:
\[ x_{vertex} = \frac{x_1 + x_2}{2} = \frac{2 + (-4)}{2} = \frac{-2}{2} = -1 \]

### Step 3: Substitute x-coordinate into the Equation
Now that we have the x-coordinate of the vertex, we can substitute \( x = -1 \) back into the original equation to find \( d \):
\[ y = a(-1 - 2)(-1 + 4) \]
Calculating the values inside the parentheses:
\[ y = a(-3)(3) \]
Thus,
\[ y = -9a \]

### Step 4: Conclusion
The y-coordinate of the vertex, \( d \), is equal to:
\[ d = -9a \]