Key Features of Graphs of Parabolas
Recall that earlier we looked at some features of lines? What might be some features of parabolas?
The General Form of the Equation for a Parabola
In order to examine the features of the parabola, let's look at the general form of the equation for a parabola. This resembles the general form of the equation for a line, as both contains variables and constants. For a parabolas, the general equation is: y = ax^2 + bx + c. All parabolas will have a fixed value for the constants a, b, and c. Let's look at some specific parabolas.
Activity: Class Discussion
What are the a,b, and c values for the following parabolas?
1. y = x^2
2. y = 3x^2 + 2x + 4
3. y = -x^2 - 3x
What would happen if a = 0?
Axis of Symmetry
All parabolas have the property of being symmetrical. Just like everything that is on the real side of a mirror appears in the virtual image created by the mirror, we can find a line of symmetry for which everything on the left side of the line has a mirror image on the other side of the line. How do we find this line? We used the formula x=-b/(2a).
Activity: Class Discussion
What is the axis of symmetry of the following equations?
1. y = x^2 + 2x + 3
2. y = x^2 + 2
3. y = 2x^2 - 4x
What other information can the axis of symmetry tell us about the parabola?
Y-intercept
The y-intercept of any parabola is the point (0,c). Note that x=0 and y=c.
Activity: Class Discussion
Why is the y-intercept of a parabola the point (0,c)? What happens to the equation of a parabola if 0 is plugged in for x?
Concavity
A parabola is said to be concave up if it extends to the left and right in an upward direction. Graph some parabolas changing the values of a. What is true about the value of a in parabolas that are concave up?
A parabola is said to be concave down it it extends to the left and right in a downward direction. Graph some parabolas changing the values of a. What is true about the value of a in parabolas that are concave down?
Reflection
What are the features of parabola graphs? What have you learned?