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

Step-by-Step Guide

1. 1

   Step 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. , The equation for parabolas that have openings facing the top and bottom use y=a(x−h)2+k{\displaystyle y=a(x-h)^{2}+k}.
   
   But if the parabola's opening faces the left or right, it will use x=a(y−h)2+k{\displaystyle x=a(y-h)^{2}+k}. , There is only one vertex per parabola.
   
   The vertex is the point on the tip of the parabola. , The x coordinate of the vertex will replace h and the y coordinate will replace k. , If the parabola is facing up , then a is positive.
   
   But if the parabola is facing down, a is negative. , Find the rise and run between this point and the vertex.
   
   Examples of coordinates with two integers are: (−4,3){\displaystyle (-4,3)}, (0,2){\displaystyle (0,2)}, and (9,9){\displaystyle (9,9)}.
   
   Examples of coordinates without two integers are: (2.5,−1){\displaystyle (2.5,-1)}, (6.97,7.1){\displaystyle (6.97,7.1)}, (−1.01,0){\displaystyle (-1.01,0)}.
   
   Remember that the rise is the difference in y and the run is the difference in x. , Note the absolute value of the run.
   
   This will be the denominator of 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 rise/run2{\displaystyle rise/run^{2}}. , This might be handy if you have to factor it out properly.
   
   If you have y=(x−4)2+4{\displaystyle y=(x-4)^{2}+4}, in standard form, it will be y=x2−8x+20{\displaystyle y=x^{2}-8x+20} which could be factored neatly into y=(x−10)(x+2){\displaystyle y=(x-10)(x+2)}. ,, Replace 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 x=ay2{\displaystyle x=ay^{2}}. , If the parabola opens to the right, a is positive.
   
   But if it opens to the left, then a is negative. , Calculate the rise and run between this point and the vertex. , Note the absolute value of the rise.
   
   This will be the denominator of 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.
2. 2

   Step 2: Remember the vertex form of a quadratic equation.

3. 3

   Step 3: Note the coordinates of the vertex.

4. 4

   Step 4: Replace h and k with the appropriate coordinates.

5. 5

   Step 5: Find out if a is positive or negative.

6. 6

   Step 6: 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).

7. 7

   Step 7: Find the value of a.

8. 8

   Step 8: Convert the equation to standard form if necessary.

9. 9

   Step 9: Remember to use the equation x=a(y−h)2+k{\displaystyle x=a(y-h)^{2}+k} since a parabola that opens sideways uses a different equation than a parabola that opens upwards or downwards.

10. 10

    Step 10: Replace h and k with the appropriate coordinates.

11. 11

    Step 11: Determine if a is positive of negative.

12. 12

    Step 12: Find the next point from the vertex on the parabola that has coordinates with two integers.

13. 13

    Step 13: Find the value of a.

Detailed Guide

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