Question

How do you find the axis of symmetry, graph and find the maximum or minimum value of the function `y = x^2 + 4x + 1`?

Answer 1

`color(green)("Minimum"->(x_("vertex")" , "y_("vertex")) -> (-2,-3))`

`color(blue)("Axis of symmetry" -> x=-2)`

Explanation:

`color(blue)("Axis of symmetry")`

This a quadratic equation and because `color(brown)(x^2" is positive")` it is of the `color(brown)("upward horse shoe shape like a letter U. ")`Thus it has a minimum value which coincides with the axis of symmetry.

As there is no coefficient in front of `x^2` the axis of symmetry and `x_("vertex")` occurs at `(-1/2)xx(+ 4)` where the 4 is from `4x`.

So the axis of symmetry is at `color(blue)(x=-2)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Determine the minimum")`

Substitute the known value for x to determine the associated value of y.

`y=(-2)^2+4(-2)+1`

`color(blue)(y=+4-8+1=-3)`
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`color(green)("Minimum"->(x_("vertex")" , "y_("vertex")) -> (-2,-3))`