Rational Functions

A rational function is a function that can be expressed as the quotient of two polynomial functions. In other words, it is a function of the form f(x) = p(x) / q(x), where p(x) and q(x) are both polynomial functions and q(x) is not the zero function.

Rational functions can have a variety of properties and behaviors, including asymptotes, intercepts, and holes. Understanding these properties is important for analyzing and graphing rational functions.

Key Concepts

Numerator and Denominator: The numerator of a rational function is the polynomial function in the numerator, and the denominator is the polynomial function in the denominator.
Vertical Asymptotes: Vertical asymptotes occur at values of x for which the denominator of the rational function is equal to zero. These are vertical lines that the graph approaches but does not cross.
Horizontal Asymptotes: Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity. The degree of the numerator and denominator polynomials determines the existence and location of horizontal asymptotes.
Intercepts: Intercepts of a rational function occur where the graph intersects the x-axis (x-intercepts) or the y-axis (y-intercepts).
Holes: Holes in the graph of a rational function occur when factors in the numerator and denominator cancel out, resulting in a gap in the graph at that point.

Study Guide

When studying rational functions, it's important to focus on the following key aspects:

Identifying the numerator and denominator of the rational function.
Finding vertical asymptotes by setting the denominator equal to zero and solving for x.
Determining horizontal asymptotes by comparing the degrees of the numerator and denominator polynomials.
Finding x-intercepts by setting the numerator equal to zero and solving for x.
Finding y-intercepts by evaluating the function at x=0.
Identifying and analyzing any holes in the graph by simplifying the function and determining where it is undefined.
Using these properties to sketch the graph of the rational function.

Practice problems involving these aspects of rational functions will help reinforce your understanding and ability to work with these types of functions.

Remember to also pay attention to any restrictions on the domain of the function, as division by zero is not allowed in the real number system.

Good luck with your studies of rational functions!

◂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.