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

Mar 21, 2016

$\textcolor{b r o w n}{\text{Axis if symmetry is the y-axis}}$
$\textcolor{b r o w n}{\text{Minimum } \to \left(x , y\right) = \left(0 , 3\right)}$

The answer can be derived without any calculations!

#### Explanation:

$\textcolor{b l u e}{\text{Determine if it is a minimum or a maximum}}$

For a quadratic; if the ${x}^{2}$ term is positive then the graph is of general shape $\cup$. On the other hand, if the ${x}^{2}$ term is negative then the graph is of general shape $\cap$

Your equation has a positive ${x}^{2}$ term. Thus we are looking for a minimum.

$\textcolor{b r o w n}{\text{The graph has a minimum value}}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{b l u e}{\text{Determine the axis of symmetry}}$

Consider the standard form equation of $y = a {x}^{2} + b x + c$

There is no term for $b x$ .

$\textcolor{b r o w n}{\text{thus the axis of symmetry is the y-axis}}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine the coordinates of the minimum value}}$

Suppose for your question there was no constant. Then the minimum would be at $\left(x , y\right) \to \left(0 , 0\right)$

The constant of 3 lifts the graph up by 3 from this point. Thus the minimum is at:

$\textcolor{b r o w n}{\text{Minimum } \to \left(x , y\right) = \left(0 , 3\right)}$