Graphs and Functions

Abstract

This lesson is designed to introduce students to graphing functions.

Objectives

Upon completion of this lesson, students will:

• have been introduced to plotting functions on the Cartesian coordinate plane
• seen several categories of functions, including lines and parabolas

Activities

This lesson introduces students to graphing functions through the following activities:

• Graph Sketcher
• Simple Plot

Standards

The activities and discussions in this lesson address the following Standards:

• Number sense, number operations and number relationships
• Patterns, relationships and functions

Key Terms

This lesson introduces students to the following terms through the included discussions:

• function
• negative numbers
• graph
• coordinate plane

Student Prerequisites

• Arithmetic: Students must be able to:
  • perform integer and fractional arithmetic
  • plot points on the Cartesian coordinate system
  • read the coordinates of a point from a graph
• Algebraic: Students must be able to:
  • work with very simple algebraic expressions
• Technological Students must be able to:
  • perform basic mouse manipulations such as point, click and drag
  • use a browser such as Netscape for experimenting with the activities

Teacher Preparation

Students will need:

• access to a browser
• pencil and graph paper
• copies of supplemental materials for the activities:
  • Graph Sketcher Activity Worksheet

Lesson Outline

These activities can be done individually or in teams of as many as four students. Allow for 2-3 hours of class time for the entire lesson if all portions are done in class.

1. Lead a discussion how functions and graphs are related.

2. Have the students try plotting points for several simple functions to ensure that they have some skill at plotting by hand. Even if graphing calculators are available, have the students plot points on graph paper - this is a skill that is important to practice by hand. Here are a few functions that might be assigned:
   1. y = 3x - 2
   2. y = x^2
   3. y = 3 - 4x
   4. y = 4 - x^2

3. Practice the students' function plotting skills by having them check their work from the previous activity by plotting the same functions using the Graph Sketcher activity.

4. Have the students investigate functions of the form y = _____ x + ____ using the Graph Sketcher activity to determine what kinds of functions come from this form, and what changing each constant does to the function. Be sure to have them keep track of what they try and record their hypotheses and observations.

   Relate these graphs to the lesson on Linear Functions to demonstrate the rationale for the terms m = slope and b = intercept in the formula Y = m * X + b.

5. Repeat the previous activity with functions of the form y = ____ x^2 + ____.

Alternate Outlines

This lesson can be rearranged in several ways.

• Replace all Graph Sketcher activities with graphing calculator activities. Note: Depending on the graphing calculator, you might have to spend some additional time discussing setting the window ranges.
• Replace all Graph Sketcher activities with Simple Plot activities. Simple Plot is a point plotting activity, which requires that the students create tables of values for the functions before plotting.
• Limit investigations to functions with one operation as in the Introduction to Functions lesson and/or to linear functions as in the Linear Functions lesson.

Extensions

After these discussions and activities, students will have more experience with functions and graphing. The next lesson, Reading Graphs, shows the students that graphs can be used to convey lots of information about a given situation.