Learning Math: Patterns, Functions, and Algebra
Nonlinear Functions Part D: Quadratic Functions (25 minutes)
Session 7, Part D
In This Part
- Quadratic Functions and Differences
Quadratic Functions and Differences
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.
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|
Fill in both the missing outputs and missing differences. Describe a pattern in the differences. Are the differences constant?
Tip: Remember, “constant” means the number remains the same: 5, 5, 5, … . A pattern may or may not be a constant pattern.
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|
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.
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)
|Difference between outputs||Second differences|
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!
Create your own quadratic function. Tabulate it and look at the differences and second differences. What seems to be true about quadratic functions?
In this session, we’ve seen some of the differences between linear functions, quadratic functions, and exponential functions and learned about their practical uses. The following Interactive Activity presents a graphical comparison between three functions that have similar equations but very different graphs:
y = 2x
y = x2
y = 2x
Which of these is an exponential function? Which is a quadratic function? Which is a linear function? For each, explain how you know.
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.
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.
Here is the completed table:
The differences are not constant, but they increase by 2 for each successive difference.
Here is the completed table:
The difference between outputs forms a linear function, and the second differences are constant (in this case, they’re always equal to 2).
Here is the completed table:
As before, the first differences form a linear function, and the second differences will always be constant.
You should find that with any quadratic function, the first differences will always form a linear function, and the second differences will always be constant.
Session 1 Algebraic Thinking
In this initial session, we will explore algebraic thinking first by developing a definition of what it means to think algebraically, then by using algebraic thinking skills to make sense of different situations.
Session 2 Patterns in Context
Explore the processes of finding, describing, explaining, and predicting using patterns. Topics covered include how to determine if patterns in tables are uniquely described and how to distinguish between closed and recursive descriptions. This session also introduces the idea that there are many different conceptions of what algebra is.
Session 3 Functions and Algorithms
In Session 1, we looked at patterns in pictures, charts, and graphs to determine how different quantities are related. In Session 2, we used patterns and variables to describe relationships in tables and in situations like toothpicks and triangles. This session extends the exploration of relationships to include the concepts of algorithm and function. Note1
Session 4 Proportional Reasoning
Look at direct variation and proportional reasoning. This investigation will help you to differentiate between relative and absolute meanings of "more" and to compare ratios without using common denominator algorithms. Topics include differentiating between additive and multiplicative processes and their effects on scale and proportionality, and interpreting graphs that represent proportional relationships or direct variation.
Session 5 Linear Functions and Slope
Explore linear relationships by looking at lines and slopes. Using computer spreadsheets, examine dynamic dependence and linear relationships and learn to recognize linear relationships expressed in tables, equations, and graphs. Also, explore the role of slope and dependent and independent variables in graphs of linear relationships, and the relationship of rates to slopes and equations.
Session 6 Solving Equations
Look at different strategies for solving equations. Topics include the different meanings attributed to the equal sign and the strengths and limitations of different models for solving equations. Explore the connection between equality and balance, and practice solving equations by balancing, working backwards, and inverting operations.
Session 7 Nonlinear Functions
Continue exploring functions and relationships with two types of non-linear functions: exponential and quadratic functions. This session reveals that exponential functions are expressed in constant ratios between successive outputs and that quadratic functions have constant second differences. Work with graphs of exponential and quadratic functions and explore exponential and quadratic functions in real-life situations.
Session 8 More Nonlinear Functions
Investigate more non-linear functions, focusing on cyclic and reciprocal functions. Become familiar with inverse proportions and cyclic functions, develop an understanding of cyclic functions as repeating outputs, work with graphs, and explore contexts where inverse proportions and cyclic functions arise. Explore situations in which more than one function may fit a particular set of data.
Session 9 Algebraic Structure
Take a closer look at "algebraic structure" by examining the properties and processes of functions. Explore important concepts in the study of algebraic structure, discover new algebraic structures, and solve equations in these new structures.
Session 10 Classroom Case Studies, Grades K-2
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the K-2 grade band.
Session 11 Classroom Case Studies, Grades 3-5
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 3-5 grade band.
Session 12 Classroom Case Studies, Grades 6-8
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 6-8 grade band.