Question

select the inequality which represents the graph below
answer
$\circ\\ 24 - y \leq -x^2 + 10x$
$\circ\\ 24 - y \geq -x^2 - 10x$
$\circ\\ 24 - y \leq -x^2 - 10x$
$\circ\\ 24 - y \geq -x^2 + 10x$

Explanation:

Step1: Find parabola roots

The parabola crosses the x-axis at $x=4$ and $x=6$.

Step2: Write factored quadratic form

Since it opens upward, the quadratic is $y = a(x-4)(x-6)$.

Step3: Solve for leading coefficient $a$

Substitute vertex $(5, -1)$:
$-1 = a(5-4)(5-6)$
$-1 = a(1)(-1)$
$a=1$
So $y=(x-4)(x-6)=x^2-10x+24$

Step4: Rearrange to match option form

$y \geq x^2-10x+24$
Rearrange: $24 - y \leq -x^2 + 10x$

Step5: Verify inequality direction

Shaded area is above the parabola, so $y$ is greater than or equal to the quadratic, which matches the rearranged inequality.

Answer:

$24 - y \leq -x^2 + 10x$