From Graphs of Parabolas to Functions of Parabolas
Creating a Function to Describe a Parabolic Graph
In Unit 3, you will be creating your own graphs. These graphs will be described by creating functions to represent them. Therefore, let's explore how to create functions that describe parabolas. Recall the general equation of a parabola: y = ax^2 + bx + c where a cannot be equal to 0. We used this general equation to find the values a and b in the formula for axis of symmetry: x = -b/(2a). To find a function which approximately fits a graph of a parabola is not a perfect science - a lot of guess and check will have to be used. This takes some courage to manipulate different aspects of the equation until you can find an equation that works just right. Making different graphs might help in the process.
Using the Axis of Symmetry Formula
If you can spot the axis of symmetry on a graph, then you can use this information to find a and b in the function of the parabola. Suppose we want to create a parabola with an axis of symmetry at x = 3. We can work backwards and choose an a and b in the axis of symmetry formula.
Choose an a and b for an axis of symmetry at x = 3.
Other Strategies
- Whether or not a is a positive or negative number determines whether or not the parabola "opens" up or down. If a is positive, the parabola's lines will extend upward - like a smile - and the parabola will have a minimum. If a is negative, the parabola's lines will extend downward - like a frown - and the parabola will have a minimum.
- Draw the parabola without the c value. You can then adjust the height of the maximum or minimum value of the parabola by adding a negative or positive c value. If c is negative, the graph will shift down by c. If c is positive the graph will shift up by c. Remember that (0,c) is the y-intercept.
- The width of the parabola can be controlled by changing a and b. If you change a and b, make sure that a and b still work in the axis of symmetry formula.
- Use the points on the graph as guidelines. If there is an ordered pair (3,9), we can plug in these x and y values in to the general equation for a parabolas and get 9 = a*3^2 + b*3 + c. In this case it must be such that a = 1, b = 0, and c = 0 so that we get 9 = 1*3^2 + 0*3 + 0 = 1*3^2 = 3^2 = 9.
Pablo's Parabolas
Pablo is a mathematician who was recently given some graphs of parabolas by another mathematician friend he is in correspondence with. Pablo wants to create functions to describe these parabolas. He quickly finds the ordered pairs of some points on the parabolas. Acivity: Group Collaboration then Class Discussion
Write a function that describe each of the following parabolas:
1. One parabola has the following points:
(-3, -5) ; (-2, 0) ; (-1, 3) ; (0, 4) ; (1, 3) ; (2, 0) ; (3, -5)
2. The other parabola has the following points:
(-3, 12) ; (-2, 4) ; (-1, 0) ; (0, 0) ; (1, 4) ; (2, 12) ; (3, 24)
Reflection
How can quadratic equations be used to describe graphs? What have you learned?