Question

1. graph: $f(x) = -2x^2 + 4x + 8$

a. what is the parent function?

Explanation:

The parent function is the simplest form of the given quadratic function, which has no transformations (no vertical/horizontal shifts, stretches, or reflections applied).

Answer:

$g(x) = x^2$

Additionally, for reference to graph $f(x) = -2x^2 + 4x + 8$:

1. Convert to vertex form:

Step1: Factor out -2 from first two terms

$f(x) = -2(x^2 - 2x) + 8$

Step2: Complete the square inside parentheses

$f(x) = -2(x^2 - 2x + 1 - 1) + 8 = -2((x-1)^2 -1) +8$

Step3: Simplify to vertex form

$f(x) = -2(x-1)^2 + 2 + 8 = -2(x-1)^2 + 10$
Vertex is $(1, 10)$, y-intercept is $(0,8)$, x-intercepts found by solving $-2x^2 +4x +8=0$:

Step4: Solve for x-intercepts

$x^2 -2x -4=0$
$x = \frac{2\pm\sqrt{4 + 16}}{2} = \frac{2\pm\sqrt{20}}{2} = 1\pm\sqrt{5} \approx 1\pm2.24$
So x-intercepts at $(1+\sqrt{5},0)$ and $(1-\sqrt{5},0)$