Question

What is the vertex form of `y=41x^2-3x+17`?

Answer 1

The wording 'vertex form' is new to me but I am assuming it is completion of the square:
`color(green)(y=41(x-3/82)^2 +16 155/164)`

Explanation:

If I am wrong about the term then perhaps I am showing you something else that may prove useful.

`color(blue)(Step1)`
Write as `y=41(x^2-3/41x)+17..............................(1)`

At the moment I can use the equals because I have not changed any of the total values on the right hand side (RHS). However, the next stage does change the value on the right so at that point I must not use the equals sign.
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`color(blue)(Step2)`
Consider the `-3/41x`. Suppose we change this to `-(1/2xx3/41)x`
We get `color(white)(...)color(blue)(3/82)color(red)(x).......................(2)`

Suppose I rewrite (1) and use the `color(blue)(3/82)` from (2)

`y->41(x-color(blue)(3/82))`

`color(green)(" Notice that the second "x " has gone and the first is no longer squared.")`

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`color(blue)(Step3)`

Now square the bracket

`y->41(x-3/82)^(color(blue)(2))`
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`color(blue)(Step4)`

Add the constant ( the right still does not `= ycolor(white)(..)` yet.

`y-> 41(x-3/82)^2 +17...................................(3)`
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`color(blue)(Step5)`
The way we have written the bracket introduces an error. Let me multiply the bracket out and you will see what I mean.

`y-> 41(x^2-6/82x+color(blue)(9/6724))+17`

`y-> (41x^2-(41xx6)/82x+color(blue)((41xx9)/(6724)))+17`

`y->41x^2-3x+color(blue)(9/164)+17`

We have introduced the extra value of `color(blue)(9/164)`

To compensate for this we remove it by subtracting the same value

`color(green)(underline("NOW IT IS = "Y))`

`y=(41x^2-3x+color(blue)(9/164))color(brown)(-9/164)+17`

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`color(blue)(Step6)`

So the vertex form (completing the square) is:
Adjust equation (3) by subtracting `color(brown)(-9/164)`

`y=41(x-3/82)^2 color(brown)(-9/164) +17`

`color(green)(y=41(x-3/82)^2 +16 155/164)`

`color(red)(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~)`