Piecewise Functions

A piecewise function is a function that is defined by several sub-functions, each corresponding to a specific interval or "piece" of the domain. These sub-functions are defined for different ranges of the independent variable.

Representation

A piecewise function can be represented using the following general form:

f(x) = { f1(x), if x < a; f2(x), if a ≤ x < b; f3(x), if x ≥ b}

Where f1(x), f2(x), f3(x) are the sub-functions and a, b are the breakpoints or transition points.

Graphing

When graphing a piecewise function, each sub-function is graphed on the corresponding interval. It's important to pay attention to the domain and range of each sub-function and the transition points between them.

Examples

Let's consider an example of a simple piecewise function:

f(x) = { 2x + 1, if x < 0; x^2, if x ≥ 0}

For x < 0, the sub-function is 2x + 1, and for x ≥ 0, the sub-function is x^2.

Study Guide

Understand the concept of piecewise functions and how they differ from regular functions.
Learn how to represent piecewise functions using mathematical notation.
Practice graphing piecewise functions and identifying the different sub-functions and their intervals.
Work on examples to understand how to evaluate piecewise functions for specific values of x.
Understand how piecewise functions are used to model real-world situations and solve problems.

It's important to practice identifying and working with piecewise functions to become comfortable with this concept. Once you understand the basics, you can explore more complex piecewise functions and their applications in mathematics and other fields.

◂Math Worksheets and Study Guides Eighth Grade. Collecting and describing data

Study Guide

Collecting and describing data

Worksheet/Answer key

Collecting and describing data

Worksheet/Answer key

Collecting and describing data

Worksheet/Answer key

Collecting and describing data

The resources above cover the following skills:

Data Analysis and Probability (NCTM)

Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.

Formulate questions, design studies, and collect data about a characteristic shared by two populations or different characteristics within one population.

Select and use appropriate statistical methods to analyze data.

Find, use, and interpret measures of center and spread, including mean and interquartile range.

Develop and evaluate inferences and predictions that are based on data.

Use conjectures to formulate new questions and plan new studies to answer them.

Grade 8 Curriculum Focal Points (NCTM)

Data Analysis and Number and Operations and Algebra: Analyzing and summarizing data sets

Students use descriptive statistics, including mean, median, and range, to summarize and compare data sets, and they organize and display data to pose and answer questions. They compare the information provided by the mean and the median and investigate the different effects that changes in data values have on these measures of center. They understand that a measure of center alone does not thoroughly describe a data set because very different data sets can share the same measure of center. Students select the mean or the median as the appropriate measure of center for a given purpose.

Connections to the Grade 8 Focal Points (NCTM)

Data Analysis: Building on their work in previous grades to organize and display data to pose and answer questions, students now see numerical data as an aggregate, which they can often summarize with one or several numbers. In addition to the median, students determine the 25th and 75th percentiles (1st and 3rd quartiles) to obtain information about the spread of data. They may use box-and-whisker plots to convey this information. Students make scatterplots to display bivariate data, and they informally estimate lines of best fit to make and test conjectures.