The standard form of a parabola is : y = ax² + bx + c. The colored points are what we will call "key points."
1) Let a>0, b =0 and c=0. What happens to the shape of the parabola as you change the value of a(be sure to keep a positive)? Now what happens to the parabola when you allow a to become negative? Describe the effect a has on the shape of the parabola.
2) As you change the values of a,b and c, what do you notice about the vertex? How would you describe the vertex in relation to the rest of the parabola? How would you describe the vertex in relation to the x intercepts?
3) Look at the pink dotted line. Why do you suppose it is called the "line of symmetry?" How does the equation of the line of symmetry relate to one of the "key points?" There is also a relationship between a,b and the line of symmetry. What is it?
4) Can you predict the y intercept just by looking at the equation in standard form?
5) y = ax² + bx + c is called a "Quadratic Function."
0 = ax² + bx + c is called a "Quadratic Equation."
What is the difference?
6) Solve the following quadratic equations:
a) x² + 2x - 3 = 0
b) -x² + 3x +4 = 0
c) 2x² - 8x =10
What is another name for "solutions to a quadratic equation?"
7) b² - 4ac is called the discriminant. On your paper, choose values for a,b and c so that b² - 4ac = 0. Graph it. Repeat using three different values for a,b and c. What do you notice about the graph when b² - 4ac = 0? If you aren't sure, try it one more time. Notice what the graphs have in common.
8)Do the same as question 7 but this time we want b² - 4ac > 0.
9) Finally, repeat problem 7 when b² - 4ac < 0.
David Cox, November 21, 2008, Created with GeoGebra