In the previous section, The Graph of the Quadratic Function, we learned the graph of a quadratic equation in general form

y = ax2 + bx + c

is a parabola.

In the following applet, you can explore what the a, b, and c variables do to the parabolic curve.

The effects of variables a and c are quite straightforward, but what does variable b do?

Things to Do

In this applet, you start with a simple quadratic curve (a parabola). You can investigate the curve as follows:

1. Use the "a" slider below the curve to vary the a parameter of the function, and see the effect on the curve.
2. Use the "c" slider below the curve to vary the c parameter of the function, and see the effect on the curve.
3. Use the "b" slider below the curve to vary the b parameter of the function.
4. Select the "Show b/2 segment" check box to see a "flipped parabola" where b=0.
5. You'll also see the value b/2, the distance from the y-axis to the (non-zero) intersection of the two parabolas, represented by a horizontal magenta (pink) segment.

The value b, of course, is twice the length of this segment.

Show b/2 segment

a
b
c

Information

[Credits: Thanks to PiPo for the idea behind this applet.]

Summary

Changing a

Varying a just changes the steepness of the arms of the curve.

Case a > 0: When a is positive, the arms of the parabola point upwards.

Case a = 0: This is a "degenerate" parabola (in this case, a straight line, parallel to the x-axis).

Case a < 0: When a is negative, the arms of the parabola point downwards.

Changing b

Changing b moves the (green) parabola along a parabolic path, given by y = -ax^2 + c (the grey parabola), and the value b is twice the length of the magenta segent (the distance from the y-axis to the intersection of the parabolas). The greater the value of b, the fuirther the green parabola moves around the grey parabola.

Case b > 0: The green parabola moves to the left and down (if a is positive).

Case b = 0: The green parabola does not move around the grey parabola in this case. The vertex will stay at (0, c).

Case b < 0: The green parabola moves to the right and down (if a is positive).

Changing c

Varying c just moves the green parabola up or down.

Case c > 0: The green parabola moves up.

Case c = 0: The green parabola does not move up or down. The vertex is at (0, 0) (if b = 0).

Case c < 0: The green parabola moves down.

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