# How do you find the axis of symmetry, graph and find the maximum or minimum value of the function y=1/2x^2?

Apr 20, 2016

color(blue)("Vertex"->(x,y)->(0,0)
color(blue)("Axis of symmetry"-> x=0)
$\textcolor{b l u e}{\text{It is a minimum}}$

#### Explanation:

$\textcolor{g r e e n}{\text{Consider each part of the standard equation "y=ax^2+bx+c" and what effect they have.}}$

color(blue)("Type 1" -> y=x^2

The coefficient of $x$ is (+1). Because this is positive the graph is of general shape $\cup$ and symmetric about the y-axis. The minimum is at $\left(x , y\right) \to \left(0 , 0\right)$

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

color(blue)("Type 2" -> y=-x^2

The coefficient of $x$ is (-1). This is negative so the graph is of general shape $\cap$ and symmetric about the y-axis. The maximum is at $\left(x , y\right) \to \left(0 , 0\right)$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Type 3" -> y=ax^2

If$\text{ "-1 < a < +1 " }$then the graph is wider (Shallower gradient )
if$\text{ "-1 > a > +1" }$then the graph narrows (Steeper gradient )

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Type 4" -> y=x^2+c

If the value of $c$ is positive it 'lifts' each point of the graph upwards by the value of $c$. Conversely if $c$ is negative it lowers the graph.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Type 5" -> y=x^2+bx+c

The coefficient of $b$ moves the plot sideways.

First write the equation as$\text{ } y = a \left({x}^{2} + \frac{b}{a} x\right) + c$

Now apply: " "(-1/2)xxb/a -> x_("vertex")
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Answering your question}}$

Given:"$\text{ } y = \frac{1}{2} {x}^{2}$

The coefficient of ${x}^{2}$ is positive so of general shape $\cup$ thus the graph has a$\textcolor{b l u e}{\text{ minimum.}}$

There is $\underline{\text{no}}$ constant value c $\underline{\text{nor}}$ is there any $b x$

So it is a mixture of Type 1 and Type 3. Consequently

color(blue)("Vertex"->(x,y)->(0,0)
color(blue)("Axis of symmetry"-> x=0)
$\textcolor{b l u e}{\text{It is a minimum}}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(green)("This graph shows what happens when you change the coefficient of "x^2" ( changing the value of "a" in "y=ax^2"))