Learn on PengiPengi Math (Grade 8)Chapter 5: Functions

Lesson 4: Comparing Functions and Identifying Nonlinearity

In this Grade 8 lesson from Pengi Math Chapter 5, students learn to compare functions represented in different forms — such as tables and graphs — by analyzing rate of change and initial value. Students also practice identifying linear versus nonlinear functions by examining equations for exponents or variables in denominators, checking tables for constant rates of change, and recognizing straight lines versus curves on graphs. Real-world scenarios are used to reinforce the distinction between constant-rate linear models and variable-rate nonlinear models.

Section 1

Comparing Linear Functions: Initial Value

Property

When comparing linear functions y=m1x+b1y = m_1x + b_1 and y=m2x+b2y = m_2x + b_2, the y-intercepts b1b_1 and b2b_2 represent the initial values or starting points of each function.
The function with the greater y-intercept has the higher starting value.

Examples

Section 2

Comparing Linear Functions: Rate of Change

Property

When comparing linear functions, the slope determines the rate of change and direction.
Functions with positive slopes increase as xx increases, while functions with negative slopes decrease as xx increases.
The absolute value of the slope indicates how quickly the function values change.

Examples

Section 3

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

Comparing Linear Functions: Initial Value

Property

When comparing linear functions y=m1x+b1y = m_1x + b_1 and y=m2x+b2y = m_2x + b_2, the y-intercepts b1b_1 and b2b_2 represent the initial values or starting points of each function.
The function with the greater y-intercept has the higher starting value.

Examples

Section 2

Comparing Linear Functions: Rate of Change

Property

When comparing linear functions, the slope determines the rate of change and direction.
Functions with positive slopes increase as xx increases, while functions with negative slopes decrease as xx increases.
The absolute value of the slope indicates how quickly the function values change.

Examples

Section 3

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.