Question

How do you use end behavior, zeros, y intercepts to sketch the graph of `f(x)=(x-4)(x-1)(x+3)`?

Answer 1

end behavior: `(-\infty, -\infty)`, `(\infty, \infty)`
x-intercepts: `(4,0)`, `(1,0)`, and `(-3,0)`.
y-intercept: `(0,12)`

Explanation:

If you were to multiply the three factors using the distributive property, you would find that this function is a third degree polynomial because the term with the dependent variable raised to highest power would be the term `x^3`.

Since the term with the dependent variable raised to the highest power has a positive coefficient `(+1)` and an odd power `(3)`, end behavior of `f(x)` when `x` becomes more negative will move in the direction of `-infty` and when `x` becomes more positive will move in the direction of `+\infty`.
end behavior: `(-\infty, -\infty)`, `(\infty, \infty)`

To find the zeros, or x-intercepts, of the function, set `f(x) = 0` and solve for `x`. Using the zero product property, we know that `f(x) = 0` when any one of the three factors, `(x-4)`, `(x-1)`, or `(x+3)` is equal to `0`. Therefore:
x-intercepts: `(4,0)`, `(1,0)`, and `(-3,0)`.

To find the y-intercept of the function, set `x = 0` and solve for `f(x)`:
`f(x) = (x - 4)(x - 1)(x + 3)`
`f(0) = ((0) - 4)((0) - 1)((0) + 3)`
`f(0) = (- 4)(- 1)(3)`
`f(0) = 12`
y-intercept: `(0,12)`