Question

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

Explanation:

Step1: Identify roots of the function

The graph crosses/touches the x-axis at $x=-5$ (a double root, since it touches and bounces) and $x=1$. So the factored form is $y=a(x+5)^2(x-1)$.

Step2: Use y-intercept to find $a$

The y-intercept is $(0,5)$. Substitute $x=0, y=5$ into the equation:
$5=a(0+5)^2(0-1)$
$5=a(25)(-1)$
$5=-25a$
Solve for $a$: $a=\frac{5}{-25}=-\frac{1}{5}$

Step3: Write the final function

Substitute $a=-\frac{1}{5}$ back into the factored form.

Answer:

$y=-\frac{1}{5}(x+5)^2(x-1)$