Question

write a function in any form that would match the graph shown below.

Explanation:

Step1: Identify x-intercepts

The graph crosses $x=-1$ and touches $x=5$ (double root). So the factored form is $y=a(x+1)(x-5)^2$.

Step2: Solve for leading coefficient a

Use y-intercept $(0,50)$:
$50 = a(0+1)(0-5)^2$
$50 = a(1)(25)$
$a = \frac{50}{25} = 2$

Step3: Expand the function

First expand $(x-5)^2$:
$(x-5)^2 = x^2 -10x +25$
Multiply by $(x+1)$:
$(x+1)(x^2 -10x +25) = x^3 -10x^2 +25x +x^2 -10x +25 = x^3 -9x^2 +15x +25$
Multiply by $a=2$:
$y=2(x^3 -9x^2 +15x +25) = x^3 -9x^2 +15x +50$

Answer:

$y = x^3 - 9x^2 + 15x + 50$