# How do you find the important parts of the equation to graph the function y=(x-2)(x+9)?

Oct 18, 2015

x-intercepts: $2$ and $\left(- 9\right)$
y-intercept: $\left(- 18\right)$
vertex: $\left(- 3 \frac{1}{2} , - 30 \frac{1}{4}\right)$

#### Explanation:

x-intercepts
the values of $x$ which cause $y$ to be $= 0$
Based on the given form: $y = \left(x - 2\right) \left(x + 9\right)$
these values are $x = 2$ and $x = - 9$

y-intercept
the value of $y$ when $x = 0$
$y = \left(0 - 2\right) \left(x + 9\right) = - 18$

vertex
Converting into vertex form: $y = m {\left(x - a\right)}^{2} + b$
$y = \left(x - 2\right) \left(x + 9\right) = {x}^{2} + 7 x - 18$
complete the square
$y = {x}^{2} + 7 x + {\left(\frac{7}{2}\right)}^{2} - 18 - {\left(\frac{7}{2}\right)}^{2}$

$\textcolor{w h i t e}{\text{X}} = 1 {\left(x - \left(- \frac{7}{2}\right)\right)}^{2} + \left(- \frac{121}{4}\right)$
which is the vertex form with vertex at $\left(- \frac{7}{2} , - \frac{121}{4}\right) = \left(- 3 \frac{1}{2} , - 30 \frac{1}{4}\right)$

We can check the graph to see that these values are reasonable:
graph{(x-2)(x+9) [-60, 60, -40, 25]}