How to Find the Equation of a Parabola from Its Graph

To find the equation of a parabola from its graph, follow these steps:

1. Identify the Vertex: Look at the graph and determine the vertex of the parabola, which is the highest or lowest point depending on whether it opens upwards or downwards. The vertex (h, k) will help in forming the equation.
2. Determine the Direction: Check if the parabola opens upwards or downwards. If it opens upwards, the coefficient of the squared term will be positive; if it opens downwards, it will be negative.
3. Find the Focus and Directrix: If the graph provides details of the focus or directrix, these can be used to calculate the equation more precisely. The focus is a point inside the parabola, while the directrix is a line outside of it.
4. Use the Vertex Form of the Parabola: The vertex form of a parabola is given by the equation
   y = a(x - h)² + k,
   where (h,k) is the vertex. You need to solve for the value of ‘a’.
5. Find Another Point on the Parabola: Identify another point (x, y) on the graph of the parabola. You can substitute this point into your equation to find the value of ‘a’.
6. Solve for ‘a’: Substitute the coordinates of the identified point and the vertex into the equation. This will allow you to calculate the value of ‘a’.
7. Write the Final Equation: Once you have the value of ‘a’, you can write the final equation of the parabola in the vertex form or convert it to standard form if needed.

For example, if the vertex is (2, 3) and another point on the parabola is (3, 4), you can substitute these into the vertex form:

4 = a(3 - 2)² + 3

This results in:

4 = a(1) + 3 
=> a = 1

Your final equation will be:

y = (x - 2)² + 3

Following these steps carefully will result in the correct equation for any parabola based on its graph.