SOLUTION: change y=1/2x^2 + 6x + 4 into the form y=a(x-h)^2+k. How do I do this algebraically?

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Question 325327: change y=1/2x^2 + 6x + 4 into the form y=a(x-h)^2+k. How do I do this algebraically?
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
formula is y = .5x^2 + 6x + 4

You need to complete the squares on it to transform it.

Double the equation on the right hand side and then multiply it by .5.

It will be the same equation only the coefficient of the x^2 term will be equal to 1.

The equation becomes:

y = .5 * (x^2 + 12x + 8)

Complete the squares on the (x^2 + 12x) part to get:

y = .5 * ( (x+6)^2 + 8 - 36)

Combine like terms to get:

y = .5 * ( (x+6)^2 - 28)

Multiply the equation out by the .5 factor to get:

y = .5 * (x+6)^2 - 14

That's your equation.

If you graph both the original equation and the revised equation, they should be identical.

The graph of y = .5x^2 + 6x + 4 is shown below:

The graph of y = .5*(x+6)^2 - 14 is shown below:

The two equations graphed together are shown below:

You can see that the graphs are identical, so the transposition is correct.

the method used is completing the squares.

The coefficient of the x^2 term has to be equal to 1.

You achieve that by multiplying the equation by a factor and dividing the equation by the same factor.

In this equation, we multiplied the equation by a factor of 2 and then divided the whole thing by the same factor.

We started with .5x^2 + 6x + 4

We multiplied that by 2 to get x^2 + 12x + 8

We then divided the whole thing by the same factor to get:

(x^2 + 12x + 8)/2 which is the same thing as:

.5 * (x^2 + 12x + 8)

It's the same equation except that the coefficient of the x^2 term is now equal to 1 which allows us to complete the squares.

To complete the square, we divide the coefficient of the x term by 2 to get 6.

The squaring factor becomes:

.5 * (x+6)^2 + 8 - 36)

We had to subtract 36 because (x+6)^2 is equal to x^2 + 12x + 36.

Since we wanted (x+6)^2 to be equal to x^2 + 12x, we had to subtract 36 from the result.

What that says is that (x+6)^2 - 36 = x^2 + 12x

If you multiply (x+6)^2 - 36, you get x^2 + 12x + 36 - 36 which becomes x^2 + 12x.

We made (x+6)^2 - 36 equivalent to x^2 + 12x.

Our equation of .5 * (x^2 + 12x + 8) became:

.5 * ( (x+6)^2 + 8 - 36 ) which became .5 * ( (x+6)^2 - 28 ) which eventually became .5 * (x+6)^2 - 14.

Anyway, completing the squares is the way to do it.

If you need to, brush up on how to complete the squares.