# 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=ax^{2}+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:

- Use the "a" slider below the curve to vary the
*a*parameter of the function, and see the effect on the curve. - Use the "c" slider below the curve to vary the
*c*parameter of the function, and see the effect on the curve. - Use the "b" slider below the curve to vary the
*b*parameter of the function. - Select the "Show
*b*/(2*a*) segment" check box to see a "flipped parabola" where `b=0`. - You'll also see the value
*b*/(2*a*), 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 (2*a*) times the length of this segment.

*a*

*b*

*c*

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## Information

The quadratic function:

[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 (2*a*) 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.

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