# How do you graph y=-x^3-x^2+5x?

Mar 21, 2017

$y = f \left(x\right) = - {x}^{3} - {x}^{2} + 5 x$

This is a cubic function with a negative $a$ value, so its end behavior is:
${\lim}_{x \to - \infty} f \left(x\right) = \infty$
${\lim}_{x \to \infty} f \left(x\right) = - \infty$

$y = - \left(x\right) \left({x}^{2} + x - 5\right)$
From the factored form above, there is a zero at $x = 0$, so $f \left(0\right) = 0$

$f \left(x\right) = - {x}^{3} - {x}^{2} + 5 x$

$f ' \left(x\right) = - 3 {x}^{2} - 2 x + 5 = - \left(x - 1\right) \left(3 x + 5\right)$
$f ' \left(x\right)$ changes sign at $x = - \frac{5}{3}$ and at $x = 1$. It changes from negative to positive at x=-5/3 and from positive to negative at x=1. This shows how the slope of $f \left(x\right)$ changes.