# What is the vertex of y=(x-4) (x+2)?

Apr 27, 2017

The vertex is $\left(1 , - 9\right)$

#### Explanation:

You have 3 options here:

Option 1

• Multiply out to get the usual form of $y = a {x}^{2} + b x + c$
• Complete the square to get vertex form: $y = a {\left(x + b\right)}^{2} + c$

Option 2

• Find the roots , the $x$-intercepts. $\left(y = 0\right)$
• The line of symmetry is halfway between, them this gives $x$
• Use $x$ to find $y$. $\left(x , y\right)$ will be the vertex.

Option 3
- Find the line of symmetry from $x = - \frac{b}{2 a}$
Then proceed as for option 2.

Let's use option 2 as the more unusual one.

Find the $x$-intercepts of the parabola:

$y = \left(x - 4\right) \left(x + 2\right) \text{ } \leftarrow$ make $y = 0$

$0 = \left(x - 4\right) \left(x + 2\right) \text{ } \rightarrow$ gives $x = \textcolor{b l u e}{4} \mathmr{and} x = \textcolor{b l u e}{- 2}$

Find the midpoint between them: $\textcolor{red}{x} = \frac{\textcolor{b l u e}{4 + \left(- 2\right)}}{2} = \textcolor{red}{1}$

Find the $y$-value using $\textcolor{red}{x = 1}$

$y = \left(\textcolor{red}{x} - 4\right) \left(\textcolor{red}{x} + 2\right) \text{ } \rightarrow \left(\textcolor{red}{1} - 4\right) \left(\textcolor{red}{1} + 2\right) = - 3 \times 3 = - 9$

The vertex is at $\left(x , y\right) = \left(1 , - 9\right)$