5. Families of Functions (A3)
Students study the symbolic and graphical forms of each function family. By recognizing the unique characteristics of each family, they can use them as tools for solving problems or for modeling real-world situations.
High School
- A3.1 Lines and Linear Functions
- Beginning Algebra Tutorials - This is a beginning Algebra web tutorial. It covers all areas of beginning Algebra. Could be used for students to access on their own or as a teacher presentation. Student and Teacher Resource.
- Direct and Partial Variations - This site contains teacher lessons plans for direct and partial variations. The concepts of direct and partial variation are introduced and compared to each other.
- Function Evaluator and Grapher - Use of this system is pretty intuitive. Enter your function f(x) in graphing calculator format (press the "example" button to see some examples). Then specify some values of x, and press "Evaluate". To see a graph, enter a range of x-values (Xmin and xMax) and press "Graph." The graph will appear in a little window called "Your Graph."
- Function Flyer - This site allows students in Algebra and Pre-Calculus to graph equations using a java applet. Students may then observe what happens to the graph of a function when the constants are changed. The site provides printable grid sheets for "pencil and paper" activities as well as discussions on each graphing activity.
- Finite Mathematics - Outstanding - This site provides students with interactive tutorials concerning functions and linear models. Specifically, functions from the numerical and algebraic viewpoints, functions from the graphical viewpoint, linear functions and models, linear regression, families of functions, and fitting functions to data. The site provides other on-line resources and the necessary math tools.
- Maths Online Gallery: Functions 2 - Students match graphs to the functions they represent using and interactive applet. Types of functions include power, rational and trigonometric.
- Parallel and Perpendicular Lines - Students are given an equation and either an x-int, y-int, or a point and have to find the equation for the new line that is either parallel or perpendicular to the given equation.
- Vertical Line Test - An interactive site where a student can create a graph in a coordinate grid and use the vertical line test to identify if a graph is a function or not.
- The Yo-Yo Problem - This site has a couple activities for students that a teacher can print that involve representing quantities with symbols and expressing their relationship with equations. In the Yo-Y-o problem, they must write and solve an equation. In the Penny Pattern, they explore another linear model with the calculator.
- A3.2 Exponential and Logarithmic Functions
- Exponentials/Logarithms - Students can experiment with exponential or logarithmic graphs and compare them to each other. A lesson plan is provided for each activity.
- Function Evaluator and Grapher - Use of this system is pretty intuitive. Enter your function f(x) in graphing calculator format (press the "example" button to see some examples). Then specify some values of x, and press "Evaluate". To see a graph, enter a range of x-values (Xmin and xMax) and press "Graph." The graph will appear in a little window called "Your Graph."
- Modeling Orbital Debris Problems - This is an excellent activity that students can do on the computer. They must use a spreadsheet program to enter data so they can make a scatter plot and get a best fit line. They can compare models they create to the actual data. They can also answer the questions with a word processor if they desire, or they can simply use a pencil and paper. Searching the Internet may be necessary to help them find answers to some of the questions. Students are exposed to linear, quadratic, and exponential function models. It also eludes to acceleration in terms of the changing rate of the deposit of orbital debris.
- Rhinos and M&M's® (Exponential Models) - In activities that use paper folding and M&M’s, students will collect data, create scatterplots, and determine algebraic models that represent their functions. Students begin this lesson by collecting data within their groups. They fold a sheet of paper and determine the area of the smallest region after each fold. Next they draw a scatterplot of their data and determine by hand an algebraic model for it. After determining this algebraic model by hand, students collect exponential decay data using M&M’s and use the graphing calculator to determine a model for this data. Both investigations allow students to explore the patterns of exponential models in tables, graphs, and symbolic form. The final activity provides an opportunity for students to apply what they know about exponential models to future populations of African Rhinos.
- Shedding Light on the Subject: Function Models of Light Decay - Presented by NCTM (Illuminations), this website provides the teacher with a four lesson unit on the decay of light as an exponential model. Teachers will appreciate the printer friendly lesson plans accompanied by student objectives, worksheets, and references. The site includes an interactive grapher and downloadable movie clips (quick time required) for students.
- The Million $ Mission - Students can read story problems about exponential growth and calculate related values. Tables are also provided.
- The Normal Distribution - This is an interactive site that allows students to change the standard deviation on a normal curve to see how it affects the graph. It will also display a variety of histograms that could fit that curve.
- A3.3 Quadratic Functions
- Direct and Partial Variations
This site contains teacher lessons plans for direct and partial variations. The concepts of direct and partial variation are introduced and compared to each other. - Exploring Linear Functions
This is a student activity that allows the student to explore how changes in the slope and y-intercept affect a linear graph. Also, what the effect would be if they are changed separately or simultaneously. - Exploring Linear Functions: Representational Relationships
This interactive web site explores linear functons. An interactive model is included. This interactive diagram shows the graph of f(x) = mx + b. Use the sliders to change the values of m and b, and observe how the graph changes. Student interactive. - Exponentials/Logarithms - Students can experiment with exponential or logarithmic graphs and compare them to each other. A lesson plan is provided for each activity.
- Function Evaluator and Grapher - Use of this system is pretty intuitive. Enter your function f(x) in graphing calculator format (press the "example" button to see some examples). Then specify some values of x, and press "Evaluate". To see a graph, enter a range of x-values (Xmin and xMax) and press "Graph." The graph will appear in a little window called "Your Graph."
- Function Flyer - This site allows students in Algebra and Pre-Calculus to graph equations using a java applet. Students may then observe what happens to the graph of a function when the constants are changed. The site provides printable grid sheets for "pencil and paper" activities as well as discussions on each graphing activity.
- Finite Mathematics - Outstanding - This site provides students with interactive tutorials concerning functions and linear models. Specifically, functions from the numerical and algebraic viewpoints, functions from the graphical viewpoint, linear functions and models, linear regression, families of functions, and fitting functions to data. The site provides other on-line resources and the necessary math tools.
- Interactive Conic Flyer - Applet that shows the changes of a parabola when the coefficients are changed. When you get to the site, you must go to the pull down menu and click on Vertical Parabola.
- Loan Calculator - Students can set the amount of the loan, the interest rate and the number of years to see how variations in these numbers can change amount of payment and interest paid over the course of the loan. They also can set the amount of the monthly payment and calculate the amount of time required to pay off the loan.
- Maths Online Gallery: Functions 2 - Students match graphs to the functions they represent using and interactive applet. Types of functions include power, rational and trigonometric.
- Direct and Partial Variations
- A3.4 Power Functions (including roots, cubics, quartics, etc.)
- Direct and Partial Variations - This site contains teacher lessons plans for direct and partial variations. The concepts of direct and partial variation are introduced and compared to each other.
- Function Flyer - This site allows students in Algebra and Pre-Calculus to graph equations using a java applet. Students may then observe what happens to the graph of a function when the constants are changed. The site provides printable grid sheets for "pencil and paper" activities as well as discussions on each graphing activity.
- Maths Online Gallery: Functions 2 - Students match graphs to the functions they represent using and interactive applet. Types of functions include power, rational and trigonometric.
- SOSMath - Informational site consisting of the inverse and composition of functions.
- A3.5 Polynomial Functions
- Equations - Tutorial on solving equations.
- Function Flyer - This site allows students in Algebra and Pre-Calculus to graph equations using a java applet. Students may then observe what happens to the graph of a function when the constants are changed. The site provides printable grid sheets for "pencil and paper" activities as well as discussions on each graphing activity.
- Polynomial Functions - In this tutorial, we define polynomial functions. We investigate some properties of polynomials including the domain, range, roots and symmetry.
- Polynomial Functions - Polynomial functions are nothing more than a sum of power functions. As a result, certain properties of polynomials are very "power-like." When many different power functions are added together, however, polynomials begin to take on unique behaviors.
- THE VOCABULARY OF POLYNOMIAL FUNCTIONS - Functions can be categorized, and the simplest type is a polynomial function.