Learn on PengiBig Ideas Math, Course 3Chapter 6: Functions

Lesson 5: Analyzing and Sketching Graphs

In this Grade 8 lesson from Big Ideas Math Course 3, Chapter 6, students learn how to analyze and sketch graphs that represent relationships between quantities without using specific numerical values on the axes. Students interpret qualitative features of graphs — such as increasing, decreasing, constant, linear, and nonlinear sections — to describe real-world situations involving speed, temperature, height, and distance over time. They also practice sketching graphs from verbal descriptions and comparing the steepness and shape of different graphs.

Section 1

Interpreting Graph Shapes

Property

The shape of a graph describes how the output variable changes.
A graph is a straight line if the rate of change (slope) is constant.
An increasing graph (positive slope) means the quantity is growing.
A decreasing graph (negative slope) means it's shrinking.
A horizontal graph (zero slope) means the quantity is constant.

Examples

  • A student walks to the library at a constant speed, studies for an hour, then walks home at the same speed. A graph of their distance from home is an increasing line, then a horizontal line, then a decreasing line.
  • Water is poured into a bucket at a steady rate, filling it. The graph of the water level versus time is a straight line with a positive slope.

Section 2

Analyzing Graph Behavior

Property

When analyzing a graph, we examine the overall behavior and relationships shown by the curve or line. This includes identifying patterns such as where the function increases or decreases, where it reaches maximum or minimum values, and how the rate of change varies across different intervals.

Examples

Section 3

Sketching a Qualitative Graph from a Story

Property

To sketch a graph from a story, translate the verbal description into a visual representation by following these steps:

  1. Identify and Label Axes: Determine the quantities to be represented on the horizontal axis (often time) and the vertical axis.
  2. Break Down the Story: Read the story and divide it into segments where the relationship between the quantities changes.
  3. Analyze Each Segment: For each segment, determine if the vertical quantity is increasing (rising line), decreasing (falling line), or constant (horizontal line).
  4. Consider the Rate: Note the rate of change. A faster rate means a steeper line. A changing rate means a curved line.
  5. Sketch and Connect: Draw a shape for each segment and connect them to form a continuous graph that tells the story.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Interpreting Graph Shapes

Property

The shape of a graph describes how the output variable changes.
A graph is a straight line if the rate of change (slope) is constant.
An increasing graph (positive slope) means the quantity is growing.
A decreasing graph (negative slope) means it's shrinking.
A horizontal graph (zero slope) means the quantity is constant.

Examples

  • A student walks to the library at a constant speed, studies for an hour, then walks home at the same speed. A graph of their distance from home is an increasing line, then a horizontal line, then a decreasing line.
  • Water is poured into a bucket at a steady rate, filling it. The graph of the water level versus time is a straight line with a positive slope.

Section 2

Analyzing Graph Behavior

Property

When analyzing a graph, we examine the overall behavior and relationships shown by the curve or line. This includes identifying patterns such as where the function increases or decreases, where it reaches maximum or minimum values, and how the rate of change varies across different intervals.

Examples

Section 3

Sketching a Qualitative Graph from a Story

Property

To sketch a graph from a story, translate the verbal description into a visual representation by following these steps:

  1. Identify and Label Axes: Determine the quantities to be represented on the horizontal axis (often time) and the vertical axis.
  2. Break Down the Story: Read the story and divide it into segments where the relationship between the quantities changes.
  3. Analyze Each Segment: For each segment, determine if the vertical quantity is increasing (rising line), decreasing (falling line), or constant (horizontal line).
  4. Consider the Rate: Note the rate of change. A faster rate means a steeper line. A changing rate means a curved line.
  5. Sketch and Connect: Draw a shape for each segment and connect them to form a continuous graph that tells the story.