Question

The expression 8x2 − 64x + 720 is used to approximate a small town's population in thousands from 1998 to 2018, where x represents the number of years since 1998. Choose the equivalent expression that is most useful for finding the year where the population was at a minimum.

8(x − 4)2 + 592
8(x − 4)2 − 592
8(x2 − 8x + 90)
8(x2 − 8x) + 90

Answer 1

A

Hope this helps!

Answer 2

• 8(x - 4)² + 592

Step-by-step explanation:

To find the equivalent expression that is most useful for finding the year where the population was at a minimum, you should transform the given equation, 8x² − 64x + 720 , to its vertex form, y = A(x - h)² + k, where the vertex is (h, k) and it would represent the minimum (if A is positive) or the maximum (if A is negative).

The easiest way to find the vertex form is by completing squares.

This is how you complete squares and get the vertex form of the equation of the parabola:

• Given: 8x² − 64x + 720

• Introduce the dependent variable: y = 8x² − 64x + 720

• Subtract 720 from both sides: y - 720 = 8x² - 64x

• Extract common factor 8 on the right side: y - 720 = 8(x² - 8x)

• Add (8/2)² inside the parenthesis on the right side and the equivalent value to the left side: y - 720 + 8(8/2)² = 8 (x² - 8x + 4²)

• Simplify: y - 720 + 128 = 8 (x² - 8x + 16)

• Factor the perfect square trinomial: y - 592 = 8 (x - 4)²

• Add 592 to both sides: y = 8(x - 4)² + 592

So, the answer is y = 8 (x - 4)² + 592