# How do you find the axis of symmetry, and the maximum or minimum value of the function y = –3(x + 7)^2 – 10?

Mar 2, 2017

$\textcolor{p u r p \le}{\text{Thus the vertex is a maximum}}$
$\textcolor{p u r p \le}{\text{Vertex} \to \left(x , y\right) = \left(- 7 , - 10\right)}$
$\textcolor{p u r p \le}{\text{Axis of symmetry } \to x = - 7}$

#### Explanation:

This is the vertex form of equation type $y = a {x}^{2} + b x + c$

If you were to square the brackets and multiply by the -3 the ${x}^{2}$ term would be $- 3 {x}^{2}$. As this is negative the graph is of general shape $\cap$. $\textcolor{p u r p \le}{\text{Thus the vertex is a maximum}}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The coordinates of the vertex can be read directly from the given equation but you will need to 'tweak' it a bit.

Given: $y = - 3 {\left(x \textcolor{red}{+ 7}\right)}^{2} \textcolor{g r e e n}{- 10}$

${x}_{\text{vertex}} = \left(- 1\right) \times \textcolor{red}{+ 7} = - 7$

${y}_{\text{vertex}} = \textcolor{g r e e n}{- 10}$

$\textcolor{p u r p \le}{\text{Vertex} \to \left(x , y\right) = \left(- 7 , - 10\right)}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The axis of symmetry passes through the vertex and is parallel to the x-axis (for a quadratic in x as is this one)

$\textcolor{p u r p \le}{\text{Axis of symmetry } \to x = - 7}$ 