Learn on PengienVision, Mathematics, Grade 8Chapter 3: Use Functions to Model Relationships

Lesson 2: Connect Representations of Functions

In this Grade 8 enVision Mathematics lesson from Chapter 3, students learn how to represent functions using tables, equations, and graphs, distinguishing between linear and nonlinear functions. Key concepts include identifying slope and y-intercept to write linear equations, applying the vertical line test to determine whether a graph represents a function, and recognizing that nonlinear functions such as A = s² produce curved rather than straight-line graphs. The lesson builds core function fluency needed throughout eighth-grade algebra.

Section 1

Representing linear relationships

Property

Relationships between variables can be represented in three different ways:

  1. A table of values displays specific data points with precise numerical values.
  2. A graph is a visual display of the data. It is easier to spot trends and describe the overall behavior of the variables from a graph.
  3. An algebraic equation is a compact summary of the model. It can be used to analyze the model and to make predictions.

Examples

  • A gym membership costs 15 dollars a month plus a 60 dollar sign-up fee. The equation is C=60+15mC = 60 + 15m, where CC is the total cost and mm is the number of months.
  • A table shows the distance a snail travels: At 1 hour, it has moved 2 feet. At 2 hours, 4 feet. At 3 hours, 6 feet. This shows a constant speed.

Section 2

The Core Difference: Constant vs. Variable Change

Property

The core difference between these two types of functions lies in their rate of change:

  • Linear Function: Has a constant rate of change (a steady slope). Its equation can always be written in the form y=mx+by = mx + b, and its graph is a straight line.
  • Nonlinear Function: Has a variable rate of change (the steepness keeps changing). Its equation cannot be written as y=mx+by = mx + b, and its graph forms a curve.

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Representing linear relationships

Property

Relationships between variables can be represented in three different ways:

  1. A table of values displays specific data points with precise numerical values.
  2. A graph is a visual display of the data. It is easier to spot trends and describe the overall behavior of the variables from a graph.
  3. An algebraic equation is a compact summary of the model. It can be used to analyze the model and to make predictions.

Examples

  • A gym membership costs 15 dollars a month plus a 60 dollar sign-up fee. The equation is C=60+15mC = 60 + 15m, where CC is the total cost and mm is the number of months.
  • A table shows the distance a snail travels: At 1 hour, it has moved 2 feet. At 2 hours, 4 feet. At 3 hours, 6 feet. This shows a constant speed.

Section 2

The Core Difference: Constant vs. Variable Change

Property

The core difference between these two types of functions lies in their rate of change:

  • Linear Function: Has a constant rate of change (a steady slope). Its equation can always be written in the form y=mx+by = mx + b, and its graph is a straight line.
  • Nonlinear Function: Has a variable rate of change (the steepness keeps changing). Its equation cannot be written as y=mx+by = mx + b, and its graph forms a curve.