The quadratic function [tex]f(x)[/tex] has roots of -2 and 6, and it passes through the point [tex](1, 15)[/tex]. What is the vertex form of the equation of [tex]f(x)[/tex]?

A. [tex]f(x) = (x - 2)^2 + 16[/tex]
B. [tex]f(x) = (x + 2)^2 + 16[/tex]
C. [tex]f(x) = -(x - 2)^2 + 16[/tex]
D. [tex]f(x) = -(x + 2)^2 + 16[/tex]



Answer :

To find the vertex form of the quadratic function \( f(x) \) that has roots -2 and 6 and passes through the point (1, 15), we follow these steps:

1. Start with the factored form of the quadratic equation:
Since the quadratic function needs to have roots at -2 and 6, it can be written in factored form as:
[tex]\[ f(x) = a(x + 2)(x - 6) \][/tex]
Here, \(a\) is a constant that we need to determine.

2. Use the point (1, 15) to solve for \(a\):
The function passes through the point (1, 15), which means when \(x = 1\), \(f(x) = 15\). Substitute \(x = 1\) and \(f(x) = 15\) into the equation:
[tex]\[ 15 = a(1 + 2)(1 - 6) \][/tex]
Simplify inside the parentheses:
[tex]\[ 15 = a(3)(-5) \][/tex]
[tex]\[ 15 = a \cdot -15 \][/tex]
Solve for \(a\):
[tex]\[ 15 = -15a \][/tex]
[tex]\[ a = -1 \][/tex]

3. Write the quadratic function with the determined \(a\) value:
Now we know that \(a = -1\), so:
[tex]\[ f(x) = -1(x + 2)(x - 6) \][/tex]

4. Expand the function to convert it to standard form:
Expand the product:
[tex]\[ f(x) = -1(x^2 - 4x - 12) \][/tex]
Distribute \(-1\):
[tex]\[ f(x) = -x^2 + 4x + 12 \][/tex]

5. Convert the function to vertex form:
To convert \(f(x) = -x^2 + 4x + 12\) to vertex form, we complete the square.

- First, factor out \(-1\) from the quadratic and linear terms:
[tex]\[ f(x) = -(x^2 - 4x) + 12 \][/tex]

- To complete the square inside the parentheses, add and subtract \(\left(\frac{4}{2}\right)^2 = 4\):
[tex]\[ f(x) = -(x^2 - 4x + 4 - 4) + 12 \][/tex]
[tex]\[ f(x) = -((x^2 - 4x + 4) - 4) + 12 \][/tex]
[tex]\[ f(x) = -((x - 2)^2 - 4) + 12 \][/tex]

- Simplify by distributing the \(-1\):
[tex]\[ f(x) = -(x - 2)^2 + 4 + 12 \][/tex]
[tex]\[ f(x) = -(x - 2)^2 + 16 \][/tex]

So, the vertex form of the quadratic equation is:
[tex]\[ f(x) = -(x - 2)^2 + 16 \][/tex]

Therefore, the correct choice is:
[tex]\[ f(x) = -(x - 2)^2 + 16 \][/tex]