Question

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

Answer 1

`color(brown)("Axis if symmetry is the y-axis")`
`color(brown)("Minimum "->(x,y)=(0,3))`

The answer can be derived without any calculations!

Explanation:

`color(blue)("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 `uu`. On the other hand, if the `x^2` term is negative then the graph is of general shape `nn`

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

`color(brown)("The graph has a minimum value")`

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

`color(blue)("Determine the axis of symmetry")`

Consider the standard form equation of `y=ax^2+bx+c`

There is no term for `bx` .

`color(brown)("thus the axis of symmetry is the y-axis")`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Determine the coordinates of the minimum value")`

Suppose for your question there was no constant. Then the minimum would be at `(x,y)->(0,0)`

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

`color(brown)("Minimum "->(x,y)=(0,3))`