Question

the equation of a parabola is $y = x^{2} + 4x + 13$. write the equation in vertex form.
write any numbers as integers or simplified proper or improper fractions.

Explanation:

Step1: Recall vertex form

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. We need to complete the square for the given quadratic equation \( y = x^2 + 4x + 13 \).

Step2: Group \(x\)-terms

Group the \(x\)-terms: \( y=(x^2 + 4x)+13 \).

Step3: Complete the square

To complete the square for \(x^2 + 4x\), take half of the coefficient of \(x\) (which is \(4\), half of it is \(2\)), square it (\(2^2 = 4\)), and add and subtract it inside the parentheses:
\( y=(x^2 + 4x + 4 - 4)+13 \).

Step4: Rewrite as perfect square

Rewrite \(x^2 + 4x + 4\) as \((x + 2)^2\):
\( y=((x + 2)^2 - 4)+13 \).

Step5: Simplify the constants

Simplify the constants: \( y=(x + 2)^2 - 4 + 13=(x + 2)^2 + 9 \).

Answer:

\( y=(x + 2)^2 + 9 \)