Writing the Equation of Parabolas
1. Determine which pattern to use (based on whether it is horizontal or vertical)
2. Substitute in h and k
3. Choose a coordinate to substitute in and solve for a.
4. Write your final equation with a, h, and k.
2. Substitute in h and k
3. Choose a coordinate to substitute in and solve for a.
4. Write your final equation with a, h, and k.
Remember the patterns:
![](/math/calculus/images/understanding_parabolas_img2.png)
Examples:
1. Find the equation of this parabola
![](/math/calculus/images/writing_the_equation_of_parabolas_img1.png)
This is a vertical parabola, so we are using the pattern
Our vertex is (5, 3), so we will substitute those numbers in for h and k:
![](/math/calculus/images/writing_the_equation_of_parabolas_img2.png)
Our vertex is (5, 3), so we will substitute those numbers in for h and k:
![](/math/calculus/images/writing_the_equation_of_parabolas_img3.png)
Now we must choose a point to substitute in. You can choose any point on the parabola except the vertex. Let's use (4, 5). We'll substitute 4 in for x and 5 for y.
![](/math/calculus/images/writing_the_equation_of_parabolas_img4.png)
![](/math/calculus/images/writing_the_equation_of_parabolas_img5.png)
Now we'll solve for a:
![](/math/calculus/images/writing_the_equation_of_parabolas_img6.png)
![](/math/calculus/images/writing_the_equation_of_parabolas_img7.png)
5 = a(1) + 3
2 = a
2 = a
To finish, we rewrite the pattern with h, k, and a:![](/math/calculus/images/writing_the_equation_of_parabolas_img8.png)
![](/math/calculus/images/writing_the_equation_of_parabolas_img8.png)
2. Find the equation of the parabola:
![](/math/calculus/images/writing_the_equation_of_parabolas_img9.png)
This is a vertical parabola, so we are using the pattern
Our vertex is (-4, -1), so we will substitute those numbers in for h and k:
![](/math/calculus/images/writing_the_equation_of_parabolas_img10.png)
Our vertex is (-4, -1), so we will substitute those numbers in for h and k:
![](/math/calculus/images/writing_the_equation_of_parabolas_img11.png)
![](/math/calculus/images/writing_the_equation_of_parabolas_img12.png)
Now we must choose a point to substitute in. You can choose any point on the parabola except the vertex. Let's use (4, 3). We'll substitute 4 in for x and 3 for y.
![](/math/calculus/images/writing_the_equation_of_parabolas_img12.png)
![](/math/calculus/images/writing_the_equation_of_parabolas_img13.png)
Now we'll solve for a:
![](/math/calculus/images/writing_the_equation_of_parabolas_img14.png)
![](/math/calculus/images/writing_the_equation_of_parabolas_img15.png)
4 = a(16) - 4
8 = 16a
1/2 = a
To finish, we rewrite the pattern with h, k, and a:![](/math/calculus/images/writing_the_equation_of_parabolas_img16.png)
8 = 16a
1/2 = a
To finish, we rewrite the pattern with h, k, and a:
![](/math/calculus/images/writing_the_equation_of_parabolas_img16.png)
Practice: Find the equation of these parabolas.
![](/math/calculus/images/writing_the_equation_of_parabolas_img17.png)
![](/math/calculus/images/writing_the_equation_of_parabolas_img18.png)
![](/math/calculus/images/writing_the_equation_of_parabolas_img19.png)
![](/math/calculus/images/writing_the_equation_of_parabolas_img20.png)
![](/math/calculus/images/writing_the_equation_of_parabolas_img21.jpg)
Answers: 1)
![](/math/calculus/images/writing_the_equation_of_parabolas_img22.png)
![](/math/calculus/images/writing_the_equation_of_parabolas_img23.png)
![](/math/calculus/images/writing_the_equation_of_parabolas_img24.png)
![](/math/calculus/images/writing_the_equation_of_parabolas_img25.png)
![](/math/calculus/images/writing_the_equation_of_parabolas_img26.png)
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