Question

the equation of a parabola is $y = x^2 - 10x + 16$. write the equation in vertex form. write any numbers as integers or simplified proper or improper fractions.

Explanation:

Step1: Recall vertex form and complete the square

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. To convert \( y = x^2 - 10x + 16 \) to vertex form, we complete the square for the \( x \)-terms.

First, take the coefficient of \( x \), which is \(-10\). Divide it by 2: \(\frac{-10}{2} = -5\). Then square it: \((-5)^2 = 25\).

Step2: Rewrite the equation with the completed square

We add and subtract 25 inside the equation to complete the square:

$$
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Since \( x^2 - 10x + 25 = (x - 5)^2 \), we substitute that in:

$$ y=(x - 5)^2 - 9 $$

Answer:

\( y=(x - 5)^2 - 9 \)