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Patterns, Functions, and Algebra
Session 7 Part A Part B Part C Part D Homework
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Notes for Session 7, Part D

Note 12

In this last part of the session, we'll look at differences between successive outputs of quadratic functions. Successive outputs do not produce constant differences (as with linear functions) or constant ratios (as with exponential functions), but the differences do have a pattern nonetheless. It turns out that for quadratic functions the differences are linear, and the second differences -- the differences of differences -- are constant.

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Note 13

Groups: Work on Problems D1-D4 with a partner. Share results and describe the functions created for Problem D4. Out of this work, a conjecture should emerge that the second differences of quadratic functions are constant.

If there's time, finish the session by looking for what the second difference tells you about the function. For a linear function, the difference between outputs is the same as the coefficient of x in the linear equation, and the same as the slope of the line.

Look at several different quadratic functions and the corresponding second differences. Notice that half of the second difference is the coefficient of the x2 term in the equation.

Looking back at the figurate numbers, for example, the square numbers had a formula y = x2. The coefficient of x2 is 1, and the second differences were constant 2s. The triangular numbers had a formula y = x2/2 + x/2. The coefficient of x2 is 1/2, and the second differences were constant 1s.

Look for the constant second differences for the pentagonal and hexagonal numbers, and use those to help find the more difficult formulas for these numbers.

Groups: Sharing results will allow you to collect information about quadratic functions. End the session by adding quadratic and exponential functions to the list of nonlinear functions started at the beginning of the session.

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