# How do you graph f(x) = -(x-2)(x + 5)?

Jul 28, 2015

By finding the extremum and the two $x$-intercepts. And plotting them.

#### Explanation:

This is a Parabola. And one way to graph Parabolas is to find three strategic points:
$\textcolor{red}{\left(1\right)}$ The extremum :

And the extremum occurs when the slope is zero. So, we solve to equation $f ' \left(x\right) = 0$

$\implies - \left(x - 2\right) \cdot 1 - \left(x + 5\right) \cdot 1 = 0$

$\implies - 2 x - 3 = 0$

$\implies x = - \frac{3}{2}$

Next plug in $x = - \frac{3}{2}$ into $f \left(x\right)$ to get the value of $y$

$y = f \left(\frac{3}{2}\right) = - \left(- \frac{3}{2} - 2\right) \left(- \frac{3}{2} + 5\right) = \left(\frac{7}{2}\right) \left(\frac{7}{2}\right) = \frac{49}{4}$

So the extremum is $\left(- \frac{3}{2} , \frac{49}{4}\right)$

$\textcolor{red}{\left(2\right)}$ The roots(the $x$-intercept) :

We solve the equation $f \left(x\right) = 0$

$\implies - \left(x - 2\right) \left(x + 5\right) = 0$

$\implies x = 2 \text{ }$ and $\text{ } x = - 5$

Hence the intercepts are : $\left(2 , 0\right)$ and $\text{ } \left(- 5 , 0\right)$

Plot these three points and link them up to obtain a sketch of the graph of $f \left(x\right)$.