Question

Write a function using the graph ​

Answer 1

Answer:

$\displaystyle f(x)=-\frac{2}{3}(x+2)(x-3)^2$

Step-by-step explanation:

The graph corresponds to a cubic function of the form:

$f(x)=a(x-p)(x-q)(x-r)$

Where p, q, and r are the zeros of f(x).

We can clearly see there are only two crossings through the x-axis. That is because one of the roots is repeated (multiple).

Thus, the roots are p=-2, q=r=3

Substituting into the function:

$f(x)=a(x+2)(x-3)(x-3)$

$f(x)=a(x+2)(x-3)^2$

The value of a can be found by using the y-intercept seen on the graph (0,-12):

$-12=a(0+2)(0+3)^2$

Operating:

$-12=18a$

Thus:

$a = -12 / 18 = -2/3$

The function is now complete:

$\mathbf{\displaystyle f(x)=-\frac{2}{3}(x+2)(x-3)^2}$