Question

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

Answer 1

Axis of symmetry is: `x=0`
Vertex is a minimum at `(x,y)->(0,-6)`

Explanation:

Consider the standard form:`" "y=ax^2+bx+c`

Given equation:`" "y=x^2-6`

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`color(brown)("Solved by understanding the effects of the parts of the formula")`
Suppose you just had `y=ax^2`

The axis of symmetry is the y-axis

If `ax^2>0` then the graph is of general shape `uu` so it has a minimum

If `ax^2<0` then the graph is of general shape `nn` so it has a maximum.

`color(blue)("In the given equation "ax^2 > 0 " so the vertex is a minimum")`
Note that `a=+1`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Suppose you had a term `bx -> y=ax^2+bx`

Then the axis of symmetry is moved from the y-axis by `(-1/2)b`

`color(blue)("The given equation does not have a "bx" term so the axis of ")`
`color(blue)("symmetry is still the y-axis.")`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The constant `c` lifts or lowers the graph so that `y_("intercept")=c`

`color(blue)("So for this equation the vertex coincides with the y-axis at ")`
`color(blue)(y=-6)`

`color(blue)(=> "Minimum "->(x,y)->(0,-6))`

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