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Change Parabola Equation to Standard Form

The equation of a parabola has the following four simplest, standard, forms:

  • `y^2 = 4ax`
  • `y^2 = -4ax`
  • `x^2 = 4ay`
  • `x^2 = -4ay`
Properties of these standard forms are discussed here.

The equation of a parabola can be changed to match one of the standard forms given above. For example, 

`y = 2x^2 + 3x + 1`

We want to change this equation so that it matches `x^2 = 4ay`. This process can be broken down to three steps:

  1. Completing the square
  2. Rearranging the equation
  3. Substituting for x and y
These steps are discussed below with the example above.

1. Completing the square


This post describes in detail the method of completing the square, although the complete steps for that process are given below:
To complete the square, we first factor out the coefficient of x squared:
`y = 2(x^2 + 3/2x + 1/2)`
Now we take the coefficient of x, which is 3/2, divide it by 2, hence we get 3/4, and then square it, which gives us (3/4)^2. The result (3/4)^2 is then added and subtracted from the expression above, as follows:
`y = 2(x^2 + 3/2x + (3/4)^2 - (3/4)^2 + 1/2)`
Now compare the highlighted part above with the formula `a^2 + 2ab + b^2 = (a + b)^2` . That is, we take the part `x^2 + 3/2x + (3/4)^2` from the above equation, and, compare it with the formula `a^2 + 2ab + b^2`. It is clear that `a` corresponds with `x` and `b` corresponds with `3/4`. Thus, we can rewrite `x^2 + 3/2x + (3/4)^2` as `(x + 3/4)^2`. So the above equation can be rewritten as
`y = 2((x + 3/4)^2 - (3/4)^2 + 1/2)`
Now we just simplify the trailing fractions
`y = 2((x + 3/4)^2 - 1/16)`
If you didn't understand the above method of completing the square properly or want to see more examples, see this post.

2. Rearranging the equation

As you may notice, on comparing the last equation with the four standard forms described at the start, it is clear that the equation we have tends to match with `x^2 = 4ay`, although not completely. In order to make it more like the standard form, we are going to rearrange the equation a bit:

First we divide by 2 on both sides. We do this in order to move all the numbers in addition or multiplication on the side of `y` in the equation.
`y/2 = (x + 3/4)^2 - 1/16`
Next we move the fraction 1/16 to the left hand side. This is done by simply adding 1/16 to both sides of the equation:
`y/2 + 1/16 = (x + 3/4)^2`
Next we factor out the coefficient of `y` from the left side of the equation. Here the coefficient of y is 1/2, so we factor out 1/2 from the left side of the equation
`1/2(y + 1/8) = (x + 3/4)^2`
This is same as:

`(x + 3/4)^2 = 1/2(y + 1/8)`

3. Substituting for x and y

As you may notice in the last equation, if it is compared with the standard form `x^2 = 4ay`, there is `x + 3/4 `in place of `x` and `y + 1/8` in place of `y` in it.

This steps involves putting a capital `X`, or any other variable you may like, in place of whatever expression is in the equation in place of what should be there for `x` in the standard form. That is, we substitute `X` for `x + 3/4`. Likewise we substitute `Y` for `y - 1/8`. Thus we get
`X^2 = 1/2Y`
Now this equation is almost completely like the standard form `x^2 = 4ay` and can be compared with it to find its vertex, focus, directrix etc.

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