Question

What is the the vertex of `y = (x - 2)^2 + 5x+4`?

Answer 1

Vertex`->(x,y)=(-1/2,color(white)(.) 31/4)`

Explanation:

Square the brackets giving:

`y=x^2-4x+4+5x+4`

`y=x^2+x+8`
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Using part of the process of completing the square ( a sort of cheat method, but permitted).

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

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

In this case `a=1`

In that we have `1x^2` (not normally written this way).

Thus `y=a(x^2+b/ax)+c" "->" "y=(x^2+x)+8`

`color(blue)(x_("vertex")->(-1/2)xx(b/a) " "->" "(-1/2)xx1 = -1/2)`
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Determine `y_("vertex")` by substitution for `x`

`y=x^2+x+8" "->" "color(blue)(y_("vertex")=(-1/2)^2-1/2+8 = 31/4)`

Tony B