Objectives of the lesson:
After this lesson, you will be able to:
Given the vertex of parabola, find an equation of a quadratic function
Given three points of a quadratic function, find the equation that defines the function
Many real world situations that model quadratic functions are data driven. What happens when you are not given the equation of a quadratic function, but instead you need to find one?
In order to obtain the equation of a quadratic function, some information must be given. Significant data points, when plotted, may suggest a quadratic relationship, but must be manipulated algebraically to obtain an equation.
Two forms of a quadratic equation: General Form and Vertex Form
When do I use each form?
When you are given the vertex and at least one point of the parabola, you generally use the vertex form.
When you are given points that lie along the parabola, you generally use the general form.
Vertex Form
Let's use a vertex that you are familiar with: (0,0).
Use the following steps to write the equation of the quadratic function that contains the vertex (0,0) and the point (2,4).
1. Plug in the vertex.
2. Simplify, if necessary.
3. Plug in x & y coordinates of the point given.
4. Solve for "a."
5. Now substitute "a" and the vertex into the vertex form.
Our final equation looks like this:
Find the equation of a quadratic function with vertex (0,0) and containing the point (4,8).
Final equation:
General Form
Given the following points on a parabola, find the equation of the quadratic function: (1,1); (2,4); (3,9).
By solving a system of three equations with three unknowns, you can obtain values for a, b, and c of the general form.
1. Plug in the coordinates for x and y into the general form. Remember y and f(x) represent the same quantity.
2. Simplify. (Remember the order of operations)
3. Repeat steps 1 & 2 for the other two points.
4. Take two equations at a time and eliminate one variable (c works well)
5. Then repeat using two equations and eliminate the same variable you eliminated in #4.
6. Take the two resulting equations and solve the system (you may use any method).
7. After finding two of the variables, select an equation to substitute the values back into.
8. Find the third variable.
9. Substitute a, b, and c back into the general equation.
3 = 3a + b>>>>>>>>>-6 = -6a - 2b
8 = 8a + 2b>>>>>>>> 8 = 8a + 2b
>>>>>>>>>>>>>>>> 2 = 2a
>>>>>>>>>>>>>>>> 1 = a
If a = 1, then 3 = 3(1) + b, so b = 0.
Now, if a = 1 and b = 0, then 1 = 1 + 0 + c, so c = 0.
By plugging in the values for a, b, and c into the general equation, we obtain the following:
Our final equation looks like:
Find the equation of a quadratic function with the given points (3,3) ;(6,12); and (9,27).
Conclusion:
You should now be familiar enough with writing quadratic equation that you will be ready for the next activity! It involves graphing using Excel and writing the equations of quadratic functions.
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