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

`"Axis of Symmetry" -> x=-2`
Minimum at `(x,y)->(-2,-5)`

Explanation:

The standard form is `y=ax^2+bx+c`

`color(blue)(underline("Determine axis of symmetry and vertex"))`

This can be written as: `a(x^2+b/ax)+c" "` ( in your case a=1)

The axis of symmetry as at `x=(-1/2)xx(b/a)`

This is also the value of `x_("vertex")-> ("maximum/minimum")`

`color(blue)(x_("vertex")" "=" "(-1/2)xx4=color(red)(-2)" "=" Axis of Symmetry")`

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Substitute for `x` to find `y_("vertex")`

`color(blue)(=>y_("vertex")=(-2)^2+4(-2)-1 = -5)`

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`color(blue)("So vertex"->(x,y)-> (-2,-5))`

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`color(blue)(underline("Determine if maximum or minimum"))`

The coefficient of `x^2` is +1. That is, it is positive. As such the graph is of general shape `uu`.

#color(blue)("Thus the vertex is that of a minimum ")#
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