Question

question 11 (mandatory) (1 point)
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 Conversion

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 (which is 9). So we add and subtract 9 inside the parentheses, but since there's a factor of -2 outside, we have to adjust: \( y = -2[(x^2 + 6x + 9) - 9] - 7 \).

Step3: Simplify the Expression

Simplify the expression: \( y = -2(x + 3)^2 + (-2)(-9) - 7 \). Calculate \( (-2)(-9) = 18 \), then \( 18 - 7 = 11 \). So \( y = -2(x + 3)^2 + 11 \).

Answer:

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