The vertex form of a parabola is y - k = a(x - h)^2 where h and k describe how many units the graph is translated (moved) either up/down or left/right depending upon whether h and k are each negative or positive. The vertex form, as its name suggests, shows the vertex and so gives the maximum or minimum point on the parabola. This is the advantage of the vertex form.

The expanded form of a parabola is y = ax^2 + bx + c where a is not 0. The advantage of this is that it is more convenient for finding the intercepts of the graph. Because each form is useful, converting from one form to the other is helpful.

Example 1:

We will be converting from vertex form to expanded form:

Convert y - 3 = 4(x + 6)^2 to expanded form.

Expand the binomial:

y - 3 = 4(x^2 + 12x + 36),

since (x + 6)^2 = x^2 + 12x + 36. I'm assuming the reader remembers the process of expanding a binomial.

y - 3 = 4x^2 + 48x + 144

y = 4x^2 + 48x + 147

Example 2:

We will be converting from expanded form to vertex form (remember, with vertex form you want to have the answer in the form of y - k = a(x - h)^2 or x - h = a(y - k)^2. The point being you want the form a(x - h)^2. This is doing the opposite of expanding the binomial, you want to 'compact' the binomial:

In order to do this one must use the algorithm called 'completing the square'. Starting with y = 4x^2 + 48x + 147 (y = ax^2 + bx + c) I will subtract the 'c' from each side since I am adding it in the equation. We want to eliminate the 'c' from the side with the variable x.

y - 147 = 4x^2 + 48x + 147 - 147 ->

y - 147 = 4x^2 + 48x

Now I must divide both sides by 4 because I need the x^2 term to have a coefficient of 1, thus:

(y - 147)/4 = x^2 + 12x

Next, I take the coefficient in front of the x term (here it's the 12) and I divide it by 2 (which results in 6). Then I take the 6 and square it - that is 6 x 6 = 36. Then I add 36 to both sides:

Now I have (y - 147)/4 + 36 = x^2 + 12x + 36. Next, factor the right side of the equal sign:

(y - 147)/4 + 36 = (x + 6)^2

Now I will multiply through by 4 to get rid of the 4 in the denominator: (this means multiplying every term by 4: 4 x (y-147)/4 = y - 147; 4 x 36 = 144; 4 times (x + 6)^2 = 4(x + 6)^2) and I get:

y - 147 + 144 = 4(x + 6)^2

adding like terms on the left side gives:

y - 3 = 4(x + 6)^2

The original equation in vertex form. Here the vertex is (-6, 3).

Note in general terms, to complete the square on x^2 + bx you first must have a coefficient of 1 in front of the x^2. Then you add (1/2b)^2.

In the example I divided by 4 to obtain a coefficient of 1 in front of x^2; then I added (1/2x12)^2 = (6)^2 = 36 to each side.

Challenges:

1) Expand y = (x + 3)^2 + 2

2) Convert this equation to vertex form and state the ordered pair of the vertex:

y = x^2 + 18x + 6

 

Work them out before you look at the answers.

 

Back to the vertex form of a parabola...