Question

The equation above represents the graph of a parabola in the \(x y\)-plane. Which of the following represents an equivalent form of the equation that includes the minimum value of \(y\) as a constant? A) \(y-28=x(x+16)\) B) \(y=x^2+2(8 x+14)\) C) \(y=x(x+16)+28\) D) \(y=(x+8)^2-36\)

Short answer

The correct option is D) \(y = (x + 8)^2 - 36\), since it represents the equation in vertex form, including the minimum value of \(y\) as a constant.

Step-by-step solution

\(y = x^2 + 16x + 28\) #Step 2: Rewrite the equation in vertex form# To rewrite the equation in vertex form, we need to complete the square. The vertex form of a quadratic is \(y = a(x-h)^2+k\).

Find the value to complete the square

To complete the square, we need to find the value that can be added and subtracted inside the square(parentheses). We will take half of the coefficient of x and square it: \(\frac{1}{2}(16)=8; \;8^2=64\)

Complete the square in the given equation

Add and subtract the value found in the previous step (64) in the equation: \(y = x^2 + 16x + 64 - 64 + 28\) Combine constant terms: \(y = x^2 + 16x + 64 - 36\) Now the equation looks like this: \(y = (x^2 + 16x + 64) - 36\)

Write the equation in vertex form

The equation in vertex form becomes: \(y = (x + 8)^2 - 36\) Comparing this form with the given options, we find that it matches option D. #Step 3: Choose the correct option# The correct option is:

D) \(y = (x + 8)^2 - 36\)