Question

How do you write the vertex form equation of the parabola `y = 2x^2 + 8x + 18`?

Answer 1

Vertex form is `y=2(x+2)^2-4+18`

Explanation:

Given:`" "y=2x^2+8x+18`...........................(1)

Let `k` be a constant

Write as:`" "y=2(x^2+color(blue)(8/2)x)+18`

From `8/2x " apply "color(blue)((1/2)xx(8/2) =color(red)( 2))`

Write the right hand side as`" "2(x^2color(red)(+2))+18`

Now take the square from `x^(color(magenta)(2)` and take it outside the bracket

Write the right hand side as`" "2(x+2)^(color(magenta)(2))+18`
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This action has produced an error so the RHS is no longer equal to y This needs to be corrected by adding the constant `k`
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`color(brown)("Correcting the equation")`

`y=2(x+2)^2+k+18`..............................(2)

Let me show you where the error comes from. Squaring the brackets and multiplying by 2 gives

`2x^2+8x+4+k+18`

Compare to the original right
`cancel(2x^2)+cancel(8x)+cancel(18) = cancel(2x^2)+cancel(8x)+4+k+cancel(18)`

The `4+k` has to become 0 for both the original and the new equation to be the same value. Thus

`k=-4`

So equation (2) becomes

`y=2(x+2)^2-4+18`

`color(blue)("Vertex form is "y=2(x+2)^2-4+18)`

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So `k` is found by squaring the constant of 2 inside of the brackets
and changing its sign. So `k = (-1)xx(+2)^2=-4`

Answer 2

`y = 2(x+2)^2 + 10`

Explanation:

The standard form of the quadratic function is `ax^2 + bx + c`

The function `y = 2x^2 + 8x + 18" is in this form "`

where a = 2 , b = 8 and c = 18

The vertex form of the equation is: `y=a(x-h)^2 + k`

where (h,k) are the coords of the vertex.

x-coord of vertex (h) = `(-b)/(2a) = (-8)/4= -2`

y-coord of vertex (k) =`2(-2)^2+8(-2) + 18 = 10`

hence (h,k) = (-2,10) and a = 2

`rArr y = 2(x+2)^2 + 10" is the vertex form "`