# How to Quickly Determine the Equation of a Parabola in Vertex Form

Three Parts:Getting StartedDetermining the Equation of a Parabola that Opens Upwards or DownwardsDetermining the Equation of a Parabola that Opens Sideways

Linear equations are pretty easy to recognise. Rise over run, that's it. But a quadratic equation is **entirely** different and harder to recognise. This guide will be able to help you recognise and determine quadratic equations from their graphed forms.

## Steps

### Part 1 Getting Started

- 1
**Select the parabola that you want to use.**The parabola should be on a graph on a coordinate plane with*x*and*y*coordinates. - 2
**Remember the vertex form of a quadratic equation.**The equation for parabolas that have openings facing the top and bottom use . But if the parabola's opening faces the left or right, it will use . - 3
**Note the coordinates of the vertex.**There is only one vertex per parabola. The vertex is the point on the tip of the parabola.

### Part 2 Determining the Equation of a Parabola that Opens Upwards or Downwards

- 1
**Replace**The*h*and*k*with the appropriate coordinates.*x*coordinate of the vertex will replace*h*and the*y*coordinate will replace*k*. - 2
**Find out if**If the parabola is facing up , then*a*is positive or negative.*a*is positive. But if the parabola is facing down,*a*is negative. - 3
**Find the next point from the vertex on the parabola that has coordinates with two integers (it doesn't matter whether it is to the left or the right).**Find the rise and run between this point and the vertex.- Examples of coordinates with two integers are: , , and .
- Examples of coordinates without two integers are: , , .
- Remember that the rise is the difference in y and the run is the difference in x.

- 4
**Find the value of**Note the absolute value of the run. This will be the denominator of*a*.*a*. To find the numerator of*a*, simply divide the rise by the run.- For example; if the rise is 2 and the run is 1, the denominator would be 1 and the numerator would be 2 divided by 1 which is 2. Thus,
*a*would be 2. - The process of finding
*a*could be simplified to .

- For example; if the rise is 2 and the run is 1, the denominator would be 1 and the numerator would be 2 divided by 1 which is 2. Thus,
- 5
**Convert the equation to standard form if necessary.**This might be handy if you have to factor it out properly.- If you have , in standard form, it will be which could be factored neatly into .

### Part 3 Determining the Equation of a Parabola that Opens Sideways

- 1
**Remember to use the equation since a parabola that opens sideways uses a different equation than a parabola that opens upwards or downwards.** - 2
**Replace**Replace*h*and*k*with the appropriate coordinates.*h*with the*y*coordinate of the vertex and*k*with the*x*coordinate.- In the example shown in the picture, the vertex is the origin, (0, 0) so there will be no
*h*and*k*, simplifying the equation to .

- In the example shown in the picture, the vertex is the origin, (0, 0) so there will be no
- 3
**Determine if**If the parabola opens to the right,*a*is positive of negative.*a*is positive. But if it opens to the left, then*a*is negative. - 4
**Find the next point from the vertex on the parabola that has coordinates with two integers.**Calculate the rise and run between this point and the vertex. - 5
**Find the value of**Note the absolute value of the rise. This will be the denominator of*a*.*a*. To find the numerator of*a*, divide the run by the rise.- If the rise is 5 and the run is 20, then
*a*will be 4/5 because we can get 4 by dividing 20 and 5. - Remember that
*a*could also be calculated by dividing the rise by the run squared. But for a parabola that opens sideways, it is run divided by the rise squared.

- If the rise is 5 and the run is 20, then

## Tips

- A good way to make sure you got the equation correct is cross referencing it with the vertex and a few other points just to make sure that the equation works fine.

## Article Info

Categories: Algebra