Questions: completing the square, then complete f(x)=-x^2-2 x+15 Rewrite the quadratic function in vertex form f(x)=

Transcript text: completing the square, then complete \[ f(x)=-x^{2}-2 x+15 \] Rewrite the quadratic function in vertex form \[ f(x)=\square \]

Solution

Solution Steps

To rewrite the quadratic function in vertex form, we need to complete the square. This involves rearranging the quadratic expression into a perfect square trinomial plus a constant. The vertex form of a quadratic function is \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.

To rewrite the quadratic function \( f(x) = -x^2 - 2x + 15 \) in vertex form, we will complete the square.

Step 1: Factor Out the Leading Coefficient

First, factor out the leading coefficient from the terms involving \( x \).

\[ f(x) = -1(x^2 + 2x) + 15 \]

Step 2: Complete the Square

To complete the square, take the coefficient of \( x \), which is 2, divide it by 2, and square it.

\[ \left(\frac{2}{2}\right)^2 = 1 \]

Add and subtract this square inside the parentheses.

\[ f(x) = -1(x^2 + 2x + 1 - 1) + 15 \]

This can be rewritten as:

\[ f(x) = -1((x + 1)^2 - 1) + 15 \]

Step 3: Simplify the Expression