Question

Consider g(x) = −x^2 + 12x − 32 Part A: Write g(x) in vertex form, identify the vertex, and determine the x-intercepts.

Answer 1

(x - 6)² = - (y - 4)

The vertex of the parabola will be (6,4).

The x-intercepts are at (8,0) and (4,0).

Explanation:

Given the equation of parabola is g(x) = y = - x² + 12x - 32 .......... (1)

Now, converting the equation to vertex form.

Here, y = - x² + 12x - 32

⇒ y = - (x² - 12x + 36) + 4

⇒ y - 4 = - (x - 6)²

⇒ (x - 6)² = - (y - 4) ........... (2) (Answer)

Now, the vertex of the parabola will be (6,4) and it opens down. (Answer)

Let the x-intercept of the parabola is at (h,0).

Hence, from equation (2) we get,

(h - 6)² = - (0 - 4) = 4

⇒ h - 6 = ± 2

⇒ h = 8 or h = 4

Therefore, the x-intercepts are at (8,0) and (4,0). (Answer)