Question

How do you find the axis of symmetry, and the maximum or minimum value of the function `y = (x+3)^2+4`?

Answer 1

There you have it the axis of symmetry and minimum is at x = -3

Explanation:

Easiest way plot it graph{( x + 3 )^ 2 + 4 [-8, 0, 0, 10]}
It about x=-3. Incidentally finding the minimum it also will give you the axis of symmetry. How do you get the minima? well take the derivative and set it equal to zero (solve for x).

The derivative of `y = (x + 3)^2 +4` is
`dy/dx = 2x+6 = 0;` the solution to this equation is `x=-3`

There you have it the axis of symmetry and minimum is at x = -3

Answer 2

Axis of symmetry is: `x=-3`

Value minimum value of the function is:`y=4`

Explanation:

Given: `" "color(brown)(y=(x+3)^2+4`

This equation is in what is known as vertex form.

`color(blue)("To find the axis of symmetry")`
Consider the 3 from `(x+3)^2`

If we multiply this by negative 1 we have:

`-1xx3=-3`

This value of -3 is the x value for the minimum which also is the axis of symmetry. In that:

`color(blue)("axis of symmetry "-> x=-3)`

This form of equation is saying that the axis of symetry is a line parallel to the y-axis and that it passes through `x=-3`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(green)("To find the value of the equation at the minimum")`

Notice they said "value of equation". This means they wish to know the value of y

`y=(x+3)^2+4` becomes

`y=((-3)+3)^2+4`

`y=0+4`

`color(green)(y=4)`