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Graphs and Functions
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Abstract

This lesson is designed to introduce students to graphing functions. 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.

Objectives

Upon completion of this lesson, students will:

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

Standards Addressed:

Textbooks Aligned:

Student Prerequisites

  • Arithmetic: Student 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 simple algebraic expressions
  • Technological: Students must be able to:
    • perform basic mouse manipulations such as point, click and drag
    • use a browser for experimenting with the activities

Teacher Preparation

Key Terms

constant functionsFunctions that stay the same no matter what the variable does are called constant functions
constantsIn math, things that do not change are called constants. The things that do change are called variables.
coordinate planeA plane with a point selected as an origin, some length selected as a unit of distance, and two perpendicular lines that intersect at the origin, with positive and negative direction selected on each line. Traditionally, the lines are called x (drawn from left to right, with positive direction to the right of the origin) and y (drawn from bottom to top, with positive direction upward of the origin). Coordinates of a point are determined by the distance of this point from the lines, and the signs of the coordinates are determined by whether the point is in the positive or in the negative direction from the origin
coordinatesA unique ordered pair of numbers that identifies a point on the coordinate plane. The first number in the ordered pair identifies the position with regard to the x-axis while the second number identifies the position on the y-axis
functionA function f of a variable x is a rule that assigns to each number x in the function's domain a single number f(x). The word "single" in this definition is very important
graphA visual representation of data that displays the relationship among variables, usually cast along x and y axes.
negative numbersNumbers less than zero. In graphing, numbers to the left of zero. Negative numbers are represented by placing a minus sign (-) in front of the number

Lesson Outline

  1. Focus and Review

    Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson. You may ask the following questions:

    • Can someone tell me what a function is?
    • Will someone give me an example of a function?
    • Will someone give me an example of something that is not a function?

  2. Objectives

    Let the students know what it is they will be doing and learning today. Say something like this:

    • Today, class, we are going to learn more about functions.
    • We are going to use the computers to learn more about functions, but please do not turn your computers on until I ask you to. I want to show you a little about this activity first.

  3. Teacher Input

    • Lead a discussion on how functions and graphs are related.

  4. Guided Practice

    • 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
    • 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 Tool.
    • Have the students investigate functions of the form y = _____ x + ____ using the Graph Sketcher Tool 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. Independent Practice

    • Have the students repeat the previous activity with functions of the form: 
      y = ____ x^2 + ____

  6. Closure

    • You may wish to bring the class back together for a discussion of the findings. Once the students have been allowed to share what they found, summarize the results of the lesson.

Alternate Outline

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 Function Machine lesson and/or to linear functions as in the Linear Functions lesson .

Suggested Follow-Up

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.

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