Math Lesson 15.1.6 - Finding the Equation of a Parabola from its Graph

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Welcome to our Math lesson on Finding the Equation of a Parabola from its Graph, this is the sixth lesson of our suite of math lessons covering the topic of Quadratic Graphs Part One, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Finding the Equation of a Parabola from its Graph

So far, we have looked at the equation of a parabola given and the task was to plot its graph by finding first the five special points described earlier in this tutorial. Now, we will deal with the reverse situation, i.e. when the graph of a parabola is given and we want to find its equation. In simpler words, now we want to use the parabola graph to find the coefficients a and b and the constant c. Again, we consider the five special points to extract any useful information from them regarding the coefficients and the constant of the parabola. Let's explain this point through an example.

Example 5

Find the equation of the parabola shown by the graph below.

Solution 5

We can easily identify all the five special points we have discussed in theory. It is not necessary to find all of them but we will find as many as needed. Thus, the coordinates of points A and B which represent the x-intercepts of the graph are A(1, 0) and B(2, 0). Therefore, the x-coordinate of the vertex V (given the segment midpoint formula) is

From here, we can write

From the graph, we can also see that the y-intercept is at C(0, 2). Therefore, we have c = 2, as the y-coordinate of the y-intercept is equal to the constant c. So far we have only partially identified the equation of the parabola as the coefficient a still unknown. We have

We can find this coefficient by substituting the coordinates of any of the point identified so far (for example of point A) in the equation above. Thus, for x = 1 and y = 0 we obtain

Therefore, the equation of the parabola shown in the figure is

or

You have reached the end of Math lesson 15.1.6 Finding the Equation of a Parabola from its Graph. There are 6 lessons in this physics tutorial covering Quadratic Graphs Part One, you can access all the lessons from this tutorial below.

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1. Parabola Equation Feedback. Helps other - Leave a rating for this parabola equation (see below)
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3. Types of Graphs Revision Notes: Quadratic Graphs Part One. Print the notes so you can revise the key points covered in the math tutorial for Quadratic Graphs Part One
4. Types of Graphs Practice Questions: Quadratic Graphs Part One. Test and improve your knowledge of Quadratic Graphs Part One with example questins and answers
5. Check your calculations for Types of Graphs questions with our excellent Types of Graphs calculators which contain full equations and calculations clearly displayed line by line. See the Types of Graphs Calculators by iCalculator™ below.
6. Continuing learning types of graphs - read our next math tutorial: Quadratic Graphs Part Two

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