# Find Equation of Quadratic Function Given by its Graph

We continue the study of Quadratic functions and here we show by an example how to find the equation of a quadratic function given by its graph.
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## E - Exercises: Find the equation of a quadratic function given its graph

As an exercise you are asked to find the equation of a quadratic function whose graph is shown in the applet and write it in the form f(x) = a x 2 + b x + c .You may also USE this applet to Find Quadratic Function Given its Graph generate as many graphs and therefore questions, as you wish. The answer is given by the same applet.

Example: Find the graph of the quadratic function f whose graph is shown below. Solution
There are several methods to answer the above question but all of them have one idea in common: you need to understand and then select the right information from the graph.
method 1:
The above graph has two x intercepts at (-3,0) and (-1,0) and a y intercept at (0,6) . The x coordinates of the x intercepts can be used to write the equation of function f as follows:
f(x) = a(x + 3)(x + 1)
We now use the y intercept f(0) = 6
6 = a(0 + 3)(0 + 1)
and solve for a to find a = 2 . The formula for the quadratic function f is given by :
f(x) = 2(x + 3)(x + 1) = 2 x
2 + 8 x + 6
method 2:
The above parabola has a vertex at (-2 , -2) and a y intercept at (0,6) . The standard (or vertex) form of a quadratic function f can be written
f(x) = a(x + 2)
2 - 2
We use the y intercept f(0) = 6
6 = a(0 + 2)
2 - 2 . Solve for a to find a = 2 . The formula for the quadratic function f is given by :
f(x) = 2(x + 2)
2 - 2 = 2 x 2 + 8 x + 6
method 3:
Since a quadratic function has the form
f(x) = a x
2 + b x + c
we need 3 points on the graph of f in order to write 3 equations and solve for a , b and c .
The following points are on the graph of f
(-3 , 0) , (-1 , 0) and (0 , 6)
point (0 , 6) gives
f(0) = 6 = a * 0
2 + b * 0 + c = c
solve for c to obtain c = 6
The two other points give two more equations
(-3 , 0) gives f(-3) = a * (-3)
2 + b * (-3) + 6
which leads to 9 a - 3 b + 6 = 0
and (-1 , 0) gives f(-3) = a (-1)
2 + b * (-1) + 6
which becomes a - b + 6 = 0
Solve the last two equations in a and b to obtain
a = 2 and b = 8 and gives
f(x) = 2 x
2 + 8 x + 6
You may now
USE this applet to Find Quadratic Function Given its Graph generate as many graphs and therefore questions, as you wish. The answer is given by the same applet.

More on quadratic functions and related topics
Find Vertex and Intercepts of Quadratic Functions - Calculator: An applet to solve calculate the vertex and x and y intercepts of the graph of a quadratic function.