# How do you find the axis of symmetry, and the maximum or minimum value of the function  G(x) = x ^ 2 - 6?

May 8, 2016

Axis of symmetry is: $x = 0$
Vertex is a minimum at $\left(x , y\right) \to \left(0 , - 6\right)$

#### Explanation:

Consider the standard form:$\text{ } y = a {x}^{2} + b x + c$

Given equation:$\text{ } y = {x}^{2} - 6$

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$\textcolor{b r o w n}{\text{Solved by understanding the effects of the parts of the formula}}$
Suppose you just had $y = a {x}^{2}$

The axis of symmetry is the y-axis

If $a {x}^{2} > 0$ then the graph is of general shape $\cup$ so it has a minimum

If $a {x}^{2} < 0$ then the graph is of general shape $\cap$ so it has a maximum.

$\textcolor{b l u e}{\text{In the given equation "ax^2 > 0 " so the vertex is a minimum}}$
Note that $a = + 1$
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Suppose you had a term $b x \to y = a {x}^{2} + b x$

Then the axis of symmetry is moved from the y-axis by $\left(- \frac{1}{2}\right) b$

$\textcolor{b l u e}{\text{The given equation does not have a "bx" term so the axis of }}$
$\textcolor{b l u e}{\text{symmetry is still the y-axis.}}$
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The constant $c$ lifts or lowers the graph so that ${y}_{\text{intercept}} = c$

$\textcolor{b l u e}{\text{So for this equation the vertex coincides with the y-axis at }}$
$\textcolor{b l u e}{y = - 6}$

$\textcolor{b l u e}{\implies \text{Minimum } \to \left(x , y\right) \to \left(0 , - 6\right)}$

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