Session 7, Part D:

In This Part: Quadratic Functions and Differences | Summary

You know that for linear functions, the difference between successive outputs is a constant. For exponential functions, the ratio between successive outputs is a constant. Is there some similar pattern for quadratic functions? The next few problems will help you decide. Note 12

Problem D1

As described at the end of Part C, quadratic functions involve squaring an input. The simplest quadratic function is simply output = (input)2. Here's the start of a table for this function, with three columns: input, output, and the difference between successive outputs:

n

n2

Difference between outputs

0

0

 1 3

1

1

2

3

4

5

6

Fill in both the missing outputs and missing differences. Describe a pattern in the differences. Are the differences constant? Note 13

 Remember, "constant" means the number remains the same: 5, 5, 5, ... . A pattern may or may not be a constant pattern.   Close Tip Remember, "constant" means the number remains the same: 5, 5, 5, ... . A pattern may or may not be a constant pattern.

Problem D2

Add a new column to your table like the one shown below. In this column, put the "differences between differences," called the second differences. What do you notice?

n

n2

Difference between outputs

Second differences

0

0

 1 3
 2

1

1

2

3

4

5

6

 Video Segment This video segment shows how to create a table of first differences and second differences in the equation y = x2. Watch this segment after you've completed Problem D2. If you get stuck on the problem, you can watch the video segment to help you. You can find this segment on the session video, approximately 15 minutes and 42 seconds after the Annenberg Media logo.

Problem D3

In Part C, you built a table for triangular numbers. One way to write the rule for triangular numbers is

output = (n2 + n) / 2.

Create a table for this rule for triangular numbers, and look for patterns in the first and second differences.

n

(n2 + n)
2

Difference between outputs

Second differences

0

0

 1 2

1

1

2

3

4

5

6

 A quadratic function is any function that can be written as... output = A(input)2 + B(input) + C A, B, and C can be any number. The only exception is that A cannot equal zero -- if A were zero, there would be no need for the input to be squared!

Problem D4

Create your own quadratic function. Tabulate it and look at the differences and second differences. What seems to be true about quadratic functions?

n

Difference between outputs

Second differences

0

1

2

3

4

5

6

 Quadratic functions are studied in detail in physics and calculus, because a quadratic function describes anything falling under the force of gravity.

 Session 7: Index | Notes | Solutions | Video