Question

graph using desmos. identify the vertex to determine the correct equation below written in vertex form. $y = 2x^2 + 12x + 18$

$\bigcirc\\ y = 2(x - 5)^2$

$\bigcirc\\ y = 2(x + 3)^2$

$\bigcirc\\ y = (2x + 12)^2 + 18$

$\bigcirc\\ y = 2(x - 3)^2$

Explanation:

Step1: Recall vertex form of quadratic

The vertex form of a quadratic is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. For \( y = 2x^2 + 12x + 18 \), we can complete the square.

Step2: Factor out the coefficient of \( x^2 \)

Factor out 2 from the first two terms: \( y = 2(x^2 + 6x) + 18 \).

Step3: Complete the square inside the parentheses

Take half of 6 (which is 3), square it (\( 3^2 = 9 \)). Add and subtract 9 inside the parentheses: \( y = 2(x^2 + 6x + 9 - 9) + 18 \).

Step4: Rewrite as a perfect square

Rewrite \( x^2 + 6x + 9 \) as \( (x + 3)^2 \): \( y = 2((x + 3)^2 - 9) + 18 \).

Step5: Distribute and simplify

Distribute the 2: \( y = 2(x + 3)^2 - 18 + 18 \), which simplifies to \( y = 2(x + 3)^2 \).

Answer:

B. \( y = 2(x + 3)^2 \)