# What are the important points needed to graph f(x) = (x + 2)(x-5)?

Jan 27, 2016

Important points:
$\textcolor{w h i t e}{\text{XXX}}$x-intercepts
$\textcolor{w h i t e}{\text{XXX}}$y-intercept
$\textcolor{w h i t e}{\text{XXX}}$vertex

#### Explanation:

The x-intercepts
These are the values of $x$ when $y$ (or in this case $f \left(x\right)$) $= 0$
$\textcolor{w h i t e}{\text{XXX}} f \left(x\right) = 0$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow \left(x + 2\right) = 0 \mathmr{and} \left(x - 5\right) = 0$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow x = - 2 \mathmr{and} x = 5$
So the x-intercepts are at $\left(- 2 , 0\right)$ and $\left(5 , 0\right)$

The y-intercept
This is the value of $y$ ($f \left(x\right)$) when $x = 0$
$\textcolor{w h i t e}{\text{XXX}} f \left(x\right) = \left(0 + 2\right) \left(0 - 5\right) = - 10$
So the y($f \left(x\right)$)-intercept is at $\left(0 , - 10\right)$

The vertex
There are several ways to find this;
I will use conversion to vertex form $f \left(x\right) = {\left(x - \textcolor{red}{a}\right)}^{2} + \textcolor{b l u e}{b}$ with vertex at $\left(\textcolor{red}{a} , \textcolor{b l u e}{b}\right)$
$\textcolor{w h i t e}{\text{XXX}} f \left(x\right) = \left(x + 2\right) \left(x - 5\right)$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow f \left(x\right) = {x}^{2} - 3 x - 10$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow f \left(x\right) = {x}^{2} - 3 x \textcolor{g r e e n}{+ {\left(\frac{3}{2}\right)}^{2}} - 10 \textcolor{g r e e n}{- {\left(\frac{3}{2}\right)}^{2}}$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow f \left(x\right) = {\left(x - \textcolor{red}{\frac{3}{2}}\right)}^{2} + \left(\textcolor{b l u e}{- \frac{49}{4}}\right)$
So the vertex is at $\left(\frac{3}{2} , - \frac{49}{4}\right)$

Here is what the graph should look like:
graph{(y-(x+2)(x-5))(x^2+(y+10)^2-0.05)((x+2)^2+y^2-0.05)((x-5)^2+y^2-0.05)((x-3/2)^2+(y+49/4)^2-0.05)=0 [-14.52, 13.96, -13.24, 1.01]}