The factors of a quadratic equation are `(x-4)` and `(x-8)` what will be the vertex of the quadratic graph?

Options

A. `-4,-2`
B. 4,6
C. 6,4
D. `-6,-4`

Answer

To find the vertex of the quadratic graph given the factors \( (x-4) \) and \( (x-8) \), we can follow these steps:

### Step 1: Form the Quadratic Equation
The factors of the quadratic equation are \( (x-4) \) and \( (x-8) \). To form the quadratic equation, we multiply these factors:
\[ f(x) = (x - 4)(x - 8) \]

### Step 2: Expand the Equation
Now, we will expand the equation:
\[ f(x) = x^2 - 8x - 4x + 32 = x^2 - 12x + 32 \]

### Step 3: Identify Coefficients
From the expanded form \( f(x) = x^2 - 12x + 32 \), we can identify the coefficients:
- \( a = 1 \)
- \( b = -12 \)
- \( c = 32 \)

### Step 4: Calculate the x-coordinate of the Vertex
The x-coordinate of the vertex of a parabola given by the equation \( ax^2 + bx + c \) can be found using the formula:
\[ x = -\frac{b}{2a} \]
Substituting the values of \( a \) and \( b \):
\[ x = -\frac{-12}{2 \cdot 1} = \frac{12}{2} = 6 \]

### Step 5: Calculate the y-coordinate of the Vertex
To find the y-coordinate of the vertex, substitute \( x = 6 \) back into the quadratic equation:
\[ f(6) = 6^2 - 12 \cdot 6 + 32 \]
Calculating this:
\[ f(6) = 36 - 72 + 32 = -4 \]

### Step 6: Write the Vertex Coordinates
Thus, the vertex of the quadratic graph is:
\[ (6, -4) \]

### Final Answer
The vertex of the quadratic graph is \( (6, -4) \).

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