22NA09_c03_s02

Lesson Overview

• Lesson Title: Characteristics of Functions

• Provider: Big Ideas Learning, LLC

Warm-Up Exercises

• Solve the following inequalities:

  • r - 5 > 4

  • h + 3 ≤ 9

• Graph the solution for each inequality.

Learning Objectives

• Learning Target: Describe characteristics of functions.

• Success Criteria:

  • Estimate intercepts of a graph of a function.

  • Approximate when a function is positive, negative, increasing, or decreasing.

  • Sketch a graph of a function from a verbal description.

Explore It: Describing Characteristics of Functions

• Function Considered: y = x³ - 3x

• Questions for Discussion: a. What do you think it means for a function to be positive, negative, increasing, or decreasing? b. Consider numbers greater than 50 and less than -50 as inputs for the function.

  • Record outputs for both pairs of inputs to evaluate function behavior.

Characteristic Analysis

• Activity:

  • How to determine when the function is positive, negative, increasing, or decreasing based on the graph.

  • Discussion on the possibility of a graph being decreasing for its entire domain yet never negative.

  • Investigate the use of technology for analyzing function characteristics.

Key Vocabulary

• Intercepts:

  • x-intercept: The x-coordinate where the graph intersects the x-axis (y = 0).

  • y-intercept: The y-coordinate where the graph intersects the y-axis (x = 0).

Estimating Intercepts: Example

• Example: Estimate intercepts from a function's graph.

  • Function a: x-intercept at approximately (1.5, 0) and y-intercept at (0, 2).

  • Function b: x-intercepts at (1, 0) and (3, 0), y-intercept at (0, 3).

  • Study Tip: Use graphs for intercept estimation and verify via substitution into the equation.

Self-Assessment

• Review and self-rate understanding of intercept estimation and function characteristics on a scale from 1 to 4.

Key Ideas: Function Characteristics

• Positive Function: Graph lies above the x-axis.

• Negative Function: Graph lies below the x-axis.

• Increasing Function: Graph moves upward as x increases.

• Decreasing Function: Graph moves downward as x increases.

• End Behavior: Behavior of the graph as x approaches +∞ or -∞.

Example: Describing Characteristics

• Function: y = -x³ + 3x²

  • Positive: x < 0 and 0 < x < 3.

  • Negative: x > 3.

  • Increasing: 0 < x < 2.

  • Decreasing: x < 0 and x > 2.

  • End Behavior: As x → -∞, y → +∞; as x → +∞, y → -∞.

Real-Life Application: Graphing Robot Hops

• Scenario: Robots hopping on an asteroid.

• Key Points:

  • Start of hop: (0, 0)

  • Maximum height: (30, 48)

  • End of hop: (60, 0)

  • Compare paths of two robots based on their hoppers and maximum heights.

Additional Questions

• Analyze and sketch a graph comparing your distance from home on a walk versus a jog at constant speed.

• Evaluate and sketch the relationship between time and height when throwing a ball straight up into the air.

Mini-Assessment

• Estimate intercepts and describe positivity, negativity, and overall increasing or decreasing nature of a given function.

• Sketch a graph based on provided characteristics.