Graphs and Functions

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

Grade 9

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Grade 10

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Eighth Grade

  • Expressions and Equations

    • Analyze and solve linear equations and pairs of simultaneous linear equations.

  • Functions

    • Define, evaluate, and compare functions.
    • Use functions to model relationships between quantities.

Functions

  • Building Functions

    • Build a function that models a relationship between two quantities
    • Build new functions from existing functions

  • Linear, Quadratic, and Exponential Models

    • Construct and compare linear, quadratic, and exponential models and solve problems

Grades 6-8

  • Algebra

    • Represent and analyze mathematical situations and structures using algebraic symbols

Grades 9-12

  • Algebra

    • Represent and analyze mathematical situations and structures using algebraic symbols
    • Understand patterns, relations, and functions
    • Use mathematical models to represent and understand quantitative relationships

Grade 8

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 5: The learner will understand and use linear relations and functions.

Introductory Mathematics

  • Algebra

    • COMPETENCY GOAL 4: The learner will understand and use linear relations and functions.
    • COMPETENCY GOAL 5: The learner will understand and use linear relations and functions.

Algebra I

  • Algebra

    • Competency Goal 4: The learner will use relations and functions to solve problems.

3rd Grade

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of numeric patterns, symbols as representations of unknown quantity, and situations showing increase over time.

4th Grade

  • Algebra

    • Standard 4-3: The student will demonstrate through the mathematical processes an understanding of numeric and nonnumeric patterns, the representation of simple mathematical relationships, and the application of procedures to find the value of an unknown.

  • Data Analysis and Probability

    • Standard 4-6: The student will demonstrate through the mathematical processes an understanding of the impact of data-collection methods, the appropriate graph for categorical or numerical data, and the analysis of possible outcomes for a simple event.

  • Geometry

    • Standard 4-4: The student will demonstrate through the mathematical processes an understanding of the relationship between two- and three-dimensional shapes, the use of transformations to determine congruency, and the representation of location and moveme
    • Standard 4-4: The student will demonstrate through the mathematical processes an understanding of the relationship between two- and three-dimensional shapes, the use of transformations to determine congruency, and the representation of location and movement within the first quadrant of a coordinate system.

7th Grade

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of proportional relationships.

8th Grade

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of equations, inequalities, and linear functions.

  • Data Analysis and Probability

    • The student will demonstrate through the mathematical processes an understanding of the relationships between two variables within one population or sample.

  • Geometry

    • The student will demonstrate through the mathematical processes an understanding of the Pythagorean theorem; the use of ordered pairs, equations, intercepts, and intersections to locate points and lines in a coordinate plane; and the effect of a dilation
    • The student will demonstrate through the mathematical processes an understanding of the Pythagorean theorem; the use of ordered pairs, equations, intercepts, and intersections to locate points and lines in a coordinate plane; and the effect of a dilation in a coordinate plane.

Elementary Algebra

  • Elementary Algebra

    • Standard EA-1: The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.
    • Standard EA-3: The student will demonstrate through the mathematical processes an understanding of relationships and functions.
    • Standard EA-4: The student will demonstrate through the mathematical processes an understanding of the procedures for writing and solving linear equations and inequalities.
    • Standard EA-5: The student will demonstrate through the mathematical processes an understanding of the graphs and characteristics of linear equations and inequalities.
    • Standard EA-6: The student will demonstrate through the mathematical processes an understanding of quadratic relationships and functions.

Intermediate Algebra

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of functions, systems of equations, and systems of linear inequalities.
    • The student will demonstrate through the mathematical processes an understanding of quadratic equations and the complex number system.
    • The student will demonstrate through the mathematical processes an understanding of algebraic expressions and nonlinear functions.

7th Grade

  • Probability and Statistics

    • 7.17 The student, given a problem situation, will collect, analyze, display, and interpret data, using a variety of graphical methods, including frequency distributions; line plots; histograms; stem-and-leaf plots; box-and-whisker plots; and scattergrams.

4th Grade

  • Geometry

    • 4.15.b The student will describe the path of shortest distance between two points on a flat surface.
    • 4.16 The student will identify and draw representations of lines that illustrate intersection, parallelism, and perpendicularity.

  • Geomety

    • 4.15.b
    • 4.16

8th Grade

  • Patterns, Functions, and Algebra

    • 8.14a The student will describe and represent relations and functions, using tables, graphs, and rules; and
    • 8.16 The student will graph a linear equation in two variables, in the coordinate plane, using a table of ordered pairs.
    • 8.14 The student will
    • 8.16 The student will graph a linear equation in two variables, in the coordinate plane, using a

Secondary

  • Algebra II

    • AII.10 The student will investigate and describe through the use of graphs the relationships between the solution of an equation, zero of a function, x-intercept of a graph, and factors of a polynomial expression.
    • AII.18 The student will identify conic sections (circle, ellipse, parabola, and hyperbola) from his/her equations. Given the equations in (h, k) form, the student will sketch graphs of conic sections, using transformations.
    • AII.20 The student will identify, create, and solve practical problems involving inverse variation and a combination of direct and inverse variations.
    • AII.10
    • AII.18
    • AII.20

Textbooks Aligned

7th

  • Module 1 - Search and Rescue

    • Section 4: Function Models
      • Reason for Alignment: The Graphs and Functions lesson is a good follow up to the Introduction to Functions lesson, also aligned with this section of the text, by building on the graphing of functions. This one goes deeper into the vocabulary and algebra of functions. This lesson may take a while if completed together in class, but some students could move through it independently in a shorter time.

8th

  • Module 3 - The Mystery of Blacktail Canyon

    • Section 2: Equations and Graphs
      • Reason for Alignment: This is a detailed lesson on graphing functions. There are discussion suggestions, vocabulary and a Graph Sketcher Activity Worksheet already made up for practice. This lesson fits with the Graphit activity.

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 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 functions

Functions that stay the same no matter what the variable does are called constant functions

constants

In math, things that do not change are called constants. The things that do change are called variables.

coordinate plane

A 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

coordinates

A 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

function

A 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

graph

A visual representation of data that displays the relationship among variables, usually cast along x and y axes.

negative numbers

Numbers 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.