Answer:
The y-intercept is (0,0)
The x-intercepts are (-3,0), (0,0), and (4,0)
Step-by-step explanation:
So we have the function:
![f(x)=-x^3+x^2+12x](https://tex.z-dn.net/?f=f%28x%29%3D-x%5E3%2Bx%5E2%2B12x)
And we want to solve for the x- and y-intercepts.
Y)
To solve for the y-intercept, recall that the y-intercept is when the graph touches the y-axis. At that point, the x values is 0. Thus, to find the x-intercept, substitute 0 for x:
![f(x)=-(0)^3+(0)^2+12(0)](https://tex.z-dn.net/?f=f%28x%29%3D-%280%29%5E3%2B%280%29%5E2%2B12%280%29)
Simplify:
![f(x)=0](https://tex.z-dn.net/?f=f%28x%29%3D0)
So, the y-intercept is (0,0)
X)
To solve for the x-intercept(s), set the function equal to 0 and solve for x:
![0=-x^3+x^2+12x](https://tex.z-dn.net/?f=0%3D-x%5E3%2Bx%5E2%2B12x)
First, factor out a negative x:
![0=-x(x^2-x-12)](https://tex.z-dn.net/?f=0%3D-x%28x%5E2-x-12%29)
Factor within the parentheses:
![0=-x(x-4)(x+3)](https://tex.z-dn.net/?f=0%3D-x%28x-4%29%28x%2B3%29)
Zero Product Property:
![-x=0\text{ or } x-4=0\text{ or } x+3=0](https://tex.z-dn.net/?f=-x%3D0%5Ctext%7B%20or%20%7D%20x-4%3D0%5Ctext%7B%20or%20%7D%20x%2B3%3D0)
Divide by -1 on the first one. Add 4 on the second one. And subtract 3 on the right:
![x=0\text{ or } x=4\text{ or } x=-3](https://tex.z-dn.net/?f=x%3D0%5Ctext%7B%20or%20%7D%20x%3D4%5Ctext%7B%20or%20%7D%20x%3D-3)
So, our x-intercepts are:
![(-3,0), (0,0), (4,0)](https://tex.z-dn.net/?f=%28-3%2C0%29%2C%20%280%2C0%29%2C%20%284%2C0%29)
And we're done :)