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

Mar 29, 2018

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

#### 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 $\left(+ 1\right)$ and an odd power $\left(3\right)$, end behavior of $f \left(x\right)$ when $x$ becomes more negative will move in the direction of $- \infty$ and when $x$ becomes more positive will move in the direction of $+ \setminus \infty$.
end behavior: $\left(- \setminus \infty , - \setminus \infty\right)$, $\left(\setminus \infty , \setminus \infty\right)$

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

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