Question

How do you find the axis of symmetry, and the maximum or minimum value of the function `y=x^2+10x+21`?

Answer 1

Using part of the process of completing the square

Axis of symmetry `-> x=-5)`
Vertex `->" minimum "->(x,y)=(-5,-4)`

Explanation:

`color(blue)("Determine axis of symmetry & " x_("vertex"))`

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

Write as `y=a(x^2+b/ax)+c`

then we have:

`x_("vertex")=" axis of symmetry"=(-1/2)xxa/b`

`color(blue)(ul(bar(|" "x_("vertex")=" axis of symmetry"=(-1/2)xx10=-5" "|))`

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`color(blue)("To determine "y_("vertex"))`

Substitute `x=-5`

`y=x^2+10x+21" "->" "y=(-5)^2+10(-5)+21`

`" "color(blue)(ul(bar(|color(white)(2/2)y_("vertex")=-4" "|)))`
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`" "color(red)("Vertex "->(x,y)=(-5,-4)`

The coefficient of `x^2->+1` as positive then the graph is of form `uu`. `color(red)("Thus the vertex is a minimum")`

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