Question

select the inequality which represents the graph shown below.
answer
\\( y - 40 \geq x^2 - 13x \\)
\\( y - 40 \leq x^2 - 13x \\)
\\( y - 40 \leq x^2 + 13x \\)
\\( y - 40 \geq x^2 + 13x \\)

Explanation:

Step1: Identify root points

Roots at $x=-8$ and $x=-5$

Step2: Write factored quadratic

$y = a(x+8)(x+5)$

Step3: Find vertex and solve for $a$

Vertex at $x=\frac{-8-5}{2}=-6.5$, $y=-2$. Substitute:
$-2 = a(-6.5+8)(-6.5+5)$
$-2 = a(1.5)(-1.5)$
$-2 = -2.25a$
$a = \frac{8}{9}$? No, simplify: $a = \frac{-2}{-2.25} = \frac{8}{9}$? No, expand standard form:
$y = (x+8)(x+5) = x^2 +13x +40$, adjust for vertex: $y = x^2 +13x +40 -2 = x^2 +13x +38$. Rearrange: $y - 40 = x^2 +13x -2$, but check inequality: shaded area is above the parabola, so $y \geq x^2 +13x +38$, rearrange to match options:
$y - 40 \geq x^2 +13x +38 -40$
$y - 40 \geq x^2 +13x -2$? No, use option structure: test point $(-6.5, 0)$ (in shaded area) in $y-40 \geq x^2+13x$:
$0-40 \geq (-6.5)^2 +13(-6.5)$
$-40 \geq 42.25 - 84.5$
$-40 \geq -42.25$, which is true.

Step4: Confirm inequality direction

Shaded region is above the parabola, so $\geq$ is correct, and the quadratic with $+13x$ matches the leftward parabola.

Answer:

$y - 40 \geq x^2 + 13x$