We have the next function
[tex]f(x)=-2x^3+10x^2+48x[/tex]
And we must find its x-intercepts and y-intercepts
1. y-intercepts:
To find the y-intercepts we need to replace x = 0 in the function and then solve it for y
So, replacing x = 0 in the function we obtain
[tex]\begin{gathered} y=-2(0)^3+10(0)^2+48(0) \\ y=0+0+0 \\ y=0 \end{gathered}[/tex]
That means, the y-intercept of the function is 0.
2. x-intercepts:
To find the y-intercepts we need to replace y = 0 in the function and then solve it for x
So, replacing y = 0 in the function we obtain
[tex]0=-2x^3+10x^2+48x[/tex]
Now, we must solve it for x:
1. we must extract the common factor -2x
[tex]0=-2x(x^2-5x-24)[/tex]
2. we must factor the polynomial inside the parentheses
[tex]0=-2x(x-8)(x+3)[/tex]
3. We must divide both sides by -2
[tex]0=x(x-8)(x+3)[/tex]
We can see that the values for x that satisfy the equality are 0, 8 and -3
That means, the x-intercepts of the function are 0, 8 and -3.
ANSWER:
y-intercept(s): 0
x-intercept(s): 0, 8, -3
