Question

write a cubic function whose graph is shown.

Explanation:

Step1: Identify root form of cubic

The x-intercepts are $x=0$, $x=2$, $x=5$, so the function has the form $y = ax(x-2)(x-5)$, where $a$ is a constant.

Step2: Solve for $a$ using $(1,4)$

Substitute $x=1$, $y=4$ into the equation:
$4 = a(1)(1-2)(1-5)$
$4 = a(1)(-1)(-4)$
$4 = 4a$
$a = \frac{4}{4} = 1$

Step3: Expand the function

First multiply $x(x-2) = x^2 - 2x$, then multiply by $(x-5)$:
$y = (x^2 - 2x)(x-5)$
$y = x^3 - 5x^2 - 2x^2 + 10x$
$y = x^3 - 7x^2 + 10x$

Answer:

$y = x^3 - 7x^2 + 10x$