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GRE Prep: Help and Review22 chapters | 185 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

The standard form equation for parabolas is one of the two ways to write parabola equations. Learn what the other one is and how it comes into play when writing standard form equations for parabolas.

The **standard form** equation for parabolas looks like your standard quadratic:

This form provides you a couple of key bits of information.

1. The first number you see, *a*, tells you whether your parabola opens up or down. If *a* is negative, it will open downwards and look like a frown. If 'it is positive, it will open up and will look like a smile. A good way to remember this is to think of the phrase, 'Be positive; don't frown.'

2. Using both *a* and *b* will give you the axis of symmetry of the parabola. The **axis of symmetry** of the parabola is the line that acts as a mirror for the parabola. The parabola on either side of the axis of symmetry is the mirror image of the other side. The formula for finding the axis of symmetry from the standard form equation is:

The other way of writing the equation for a parabola is the **vertex form**. The vertex form gives you three bits of information about the parabola.

- Just like the standard form, the first number,
*a*, tells you whether the parabola opens up or down. If it's positive, it opens up and if it's negative, it opens down. - The number
*h*gives you the axis of symmetry,*x*=*h*. - The vertex form also gives you the vertex or tip of the parabola, (
*h*,*k*).

Now that we've covered our definitions, let's see how this works. We are given a parabola with a vertex of (1, 2) and another point (0, 5), and we're asked to find the standard form equation. What do we do?

We can't figure out the standard form directly using this information. This is where the vertex form comes in. In order to write the standard form equation, we first need to write the equation in vertex form because the information provided allows us to completely figure out the parabola's equation using that form.

The coordinates of the vertex are (1, 2), which correspond to the values (*h, k*). So we know that in this case, *h* = 1 and *k* = 2.

We can plug those values into our vertex form equation for *h* and *k*.

The only value we still need to find is *a*, but how do we do that? We were given the coordinates of the vertex, which we already used, and the coordinates of another point on the parabola (0, 5). We know that those coordinates are the *x*- and *y*-values of the parabola at that point, so let's plug in *x* = 0 and *y* = 5, and then solve the entire equation for *a*.

Once we've plugged in our *x*- and *y*-values, we can simplify what's in the parentheses, take the square, and subtract the 2 from both sides to end up with a value of 3 for *a*. We now have our quadratic equation in vertex form.

The next step is to make the conversion from vertex form to standard form.

Remember that our standard form equation for a parabola will be a standard quadratic:

Once we have the vertex form, all we need to convert it to standard form is a basic understanding of the order of operations. To begin this conversion, we start with the parentheses and multiply out the part that is squared. Using FOIL or double-distribution, we can see that (*x* - 1)^2 becomes *x*^2 - 2*x* + 1.

Next, we have to multiply through by 3 to get rid of those parentheses. From there, it's just a matter of combining our like terms and rearranging them so that they're in the order necessary for standard form.

We now have our standard form equation. Notice that the *a* for the vertex form is the same as the *a* for the standard form. This will always be true; if you ever end up with a different value for *a*, it's time to go back and check your work!

Don't forget that this equation describes a parabola, so what can we say about this parabola given this equation? First, we see that *a* = 3. That's greater than zero, so this parabola opens up and looks like a smile.

Next, we can find the axis of symmetry by using our formula:

In this case, *x* = - (-6) / 2(3), which equals 1. This means that this parabola's axis of symmetry is the line *x* = 1.

You can see that it's not difficult to describe the parabola's basic properties when it's in standard form. Converting to standard form is a two-step process, but it's fairly straightforward. All you have to do is remember the two steps. Step 1 is the writing of the vertex form, and step 2 is the conversion from vertex form to standard form.

For parabolas, we have two different ways of writing the equation. We have the **standard form** and the **vertex form**. In order to write a parabolic equation in standard form, we first need the vertex form. Most problems provide a vertex and another point so we can find the vertex form. Afterwards, we use the vertex form and convert it to the standard form.

When you are finished, you should be able to:

- Identify the vertex and standard form equations for a parabola
- Convert the vertex form equation of a parabola into standard form

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GRE Prep: Help and Review22 chapters | 185 lessons

- Multiplying Binomials Using FOIL and the Area Method 7:26
- Multiplying Binomials Using FOIL & the Area Method: Practice Problems 5:46
- How to Use the Quadratic Formula to Solve a Quadratic Equation 9:20
- How to Solve a Quadratic Equation by Factoring 7:53
- How to Solve Quadratics That Are Not in Standard Form 6:14
- What is a Parabola? 4:36
- Parabolas in Standard, Intercept, and Vertex Form 6:15
- How to Complete the Square 8:43
- Completing the Square Practice Problems 7:31
- Factoring Quadratic Equations: Polynomial Problems with a Non-1 Leading Coefficient 7:35
- How to Factor Quadratic Equations: FOIL in Reverse 8:50
- Writing Standard-Form Equations for Parabolas: Definition & Explanation 6:06
- Go to GRE - FOIL, Parabolas & Quadratics: Help & Review

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