Questions: Write the function in vertex form f(x)=x^2+8x+12

Solution Steps

The given quadratic function is in the standard form:

\[ f(x) = ax^2 + bx + c \]

For the function \( f(x) = x^2 + 8x + 12 \), we have:

• \( a = 1 \)
• \( b = 8 \)
• \( c = 12 \)

To convert the quadratic function into vertex form, we need to complete the square. The vertex form of a quadratic function is:

\[ f(x) = a(x - h)^2 + k \]

First, focus on the quadratic and linear terms: \( x^2 + 8x \).

To complete the square, take half of the coefficient of \( x \), square it, and add and subtract it inside the function:

1. Half of 8 is 4.
2. Squaring 4 gives 16.

Add and subtract 16 inside the function:

\[ f(x) = (x^2 + 8x + 16) - 16 + 12 \]

Now, simplify the expression:

\[ f(x) = (x + 4)^2 - 4 \]

This is the vertex form of the quadratic function.

Final Answer