Search IntMath
Close

450+ Math Lessons written by Math Professors and Teachers

5 Million+ Students Helped Each Year

1200+ Articles Written by Math Educators and Enthusiasts

Simplifying and Teaching Math for Over 23 Years

#### Tips, tricks, lessons, and tutoring to help reduce test anxiety and move to the top of the class.

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.

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, 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.

## Problem Solver This tool combines the power of mathematical computation engine that excels at solving mathematical formulas with the power of GPT large language models to parse and generate natural language. This creates math problem solver thats more accurate than ChatGPT, more flexible than a calculator, and faster answers than a human tutor. Learn More.