Question

question 11 (mandatory) (1 point) saved
which equation represents $y = -2x^2 - 12x - 7$ in vertex form?

a) $y = -2(x + 3)^2 + 11$

b) $y = -2(x + 3)^2 - 16$

c) $y = -2(x - 3)^2 + 11$

d) $y = -2(x + 3)^2 + 2$

Explanation:

Step1: Recall Vertex Form Method

To convert \( y = ax^2 + bx + c \) to vertex form \( y = a(x - h)^2 + k \), we complete the square. Start with \( y = -2x^2 - 12x - 7 \). Factor out the coefficient of \( x^2 \) from the first two terms: \( y = -2(x^2 + 6x) - 7 \).

Step2: Complete the Square Inside the Parentheses

For \( x^2 + 6x \), take half of 6 (which is 3), square it (\( 3^2 = 9 \)). Add and subtract this inside the parentheses: \( y = -2(x^2 + 6x + 9 - 9) - 7 \).

Step3: Rewrite as a Perfect Square

Rewrite \( x^2 + 6x + 9 \) as \( (x + 3)^2 \): \( y = -2((x + 3)^2 - 9) - 7 \). Distribute the -2: \( y = -2(x + 3)^2 + 18 - 7 \).

Step4: Simplify the Constant Term

Simplify \( 18 - 7 = 11 \), so \( y = -2(x + 3)^2 + 11 \).

Answer:

a) \( y = -2(x + 3)^2 + 11 \)