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Converting a Quadratic Equation into Vertex Form

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Today we're going to talk about converting a quadratic equation into vertex form. Vertex form is useful because by looking at a quadratic equation written in vertex form, you can easily identify the x and y-coordinates of the vertex. So we're going to start by reviewing vertex form and then we'll do an example showing how to convert a quadratic equation into vertex form.

So let's start by reviewing vertex form of a quadratic equation. The vertex form of a quadratic equation looks like this and if we were to graph an equation that was written in vertex form, it would create a parabola. And the vertex of that parabola could be easily seen from the equation in vertex form by looking at the h value, or the value for h, which would give us the x-coordinate of the vertex and the value for k, which would give us the y-coordinate of the vertex. So that's why we call this vertex form.

So the vertex, as you remember, is either the maximum point or the minimum point of the parabola. It'll be a maximum point if the parabola is upside down and it will be a minimum point if the parabola is right side up, like a "u" shape. And when we are graphing a parabola from an equation that's written in vertex form, we can simply start by graphing the vertex. And because the vertex also lies on the line of symmetry-- or the axis of symmetry for the parabola, we can plot one side of the parabola from the vertex and then simply reflect those points over the axis of symmetry to plot the points for the other side of the parabola.

So before we do examples of converting a quadratic equation to vertex form, let's compare the vertex form of a quadratic equation to the standard form. So, as we just talked about, vertex form looks like this. So for example, we could have y is equal to 3 times x minus 2 squared minus 4, and so we notice that in vertex form, we have a binomial squared. For standard form, we have y is equal to ax squared plus bx plus c. So for example, y equals 3x squared minus 12x plus 8 is written in standard form. And we notice that in standard form, we have all of our terms of the quadratic expanded. So in other words, this is written in expanded form. So when we want to write a quadratic equation from standard form to vertex form, we can remember that, when using completing the square, you are taking terms that are expanded and rewriting them so that they contain a binomial squared.

So let's do some examples to see how we can use completing the square to write some quadratic equations in vertex form. All right, so let's do an example converting a quadratic equation from standard form to vertex form. So I have y is equal to 2x squared plus 8x minus 24 and I want to write that in vertex form, so I'm going use completing the square to do that. So I'm going to start by moving this a constant term to the other side. So I'm going to add 24 to both sides of my equation. That will give me a y plus 24 equals 2x squared plus 8x.

Then I'm going to factor out of my coefficient that's in front of my x squared term from both terms on this side. So this will give me 2 times x squared plus 4x because when I distribute 2 times 4x, that would give me 8x. The other side will stay the same.

Next I'm going to use the completing the square method. So I'm going to take the coefficient in front of my x term, which is 4, and I'm going to divide that by 2, which would give me 2. Then, if I take 2 and square it, that will bring me to 4. So I know that to write this as a perfect square trinomial, I need to have a constant term of 4 here. So I'm going write that as x squared plus 4x plus 4.

To keep my equation balanced because I added a 4 here-- which is really 2 times 4 because of the 2 on the outside of the parentheses which would give me 8. Since I added value of 8 to this side of the equation, I need to add 8 to this side of the equation. So now I have y plus 24 plus 8.

So because this is a perfect square trinomial, I know that I can write it that is x plus or minus the number quantity squared. So because the value of 2 plus 2 is 4 and 2 times 2 is 4, I know that this will be x plus 2 quantity squared. And I brought my 2 down in front. On the other side I can go ahead and simplify 24 plus 8 will give me 32. And I still have my y.

Now I'm almost there, I'm almost to vertex form. I just need to bring this constant term back to the other side. So I'm gonna subtract 32 from both sides. This will give me y is equal to 2 times x plus 2 squared minus 32, which is in vertex form of this original quadratic equation that was written in standard form.

So let's go over a key points from today. The vertex of a parabola can be readily identified in the vertex form of a quadratic equation, because h and k form the x and y-coordinates of the vertex. The vertex of a parabola is the minimum point-- in the case of an upward facing parabola-- or the maximum point-- in the case of a downward facing parabola, and also lies on the parabola's axis of symmetry. And completing the square is used to write a quadratic expression in expanded form into binomial squared. This is useful when writing a quadratic equation in vertex form.

So I hope that these key points and examples helped you understand a little bit more about converting quadratic equations into vertex form. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.

Formulas to Know

- Standard Form of a Quadratic Equation
- Vertex Form of a Quadratic Equation