Interactive Quadratic Function Graph

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/(2a) segment" check box to see a "flipped parabola" where `b=0`.
5. You'll also see the value b/(2a), 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 (2a) times the length of this segment.

[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, whose slope depends on the value of b).

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 (2a) times 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) from its "normal" position with the vertex at the origin.

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 from its "normal" position with the vertex at the origin.

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.