1.1 Change in Tandem

Introduction

• Course by Mr. Kelly from FlipMath.com.

• Lesson 1.1: "Change in Tandem".

• Focus on reviewing fundamental concepts of functions.

Functions

• Definition: A function is a mathematical relation that maps each input value to exactly one output value.

  • Input: Known as domain or independent variable (typically denoted by X).

  • Output: Known as range or dependent variable (commonly denoted by Y).

  • In AP Statistics, output is referred to as the response variable.

• Graphing: Important to label axes correctly with input (X) on the horizontal and output (Y) on the vertical.

Variable Representation

• Variables can be represented differently based on the context of the problem.

• Example Variables:

  • H: Height of a football.

  • T: Seconds since the football was thrown.

• Function Notation: In function notation, the input (T) goes inside parentheses, output (H) is shown as H(T).

Example: Water in a Pool

• Scenario: Function W(X) where W represents the amount of water in gallons and X represents the length of the pool.

  • Independent Variable (X): Length of the pool.

  • Dependent Variable (W): Amount of water in the pool.

Varying in Tandem

• Changes in independent and dependent variables are described by a function rule.

• Methods of expressing function rules:

  • Verbally: Describing the relationship in words.

  • Analyzer: Using mathematical reasoning.

  • Numerically: Utilizing tables of values.

  • Graphically: Plotting the relationship on a graph.

• Mnemonic: VANG (Verbal, Analytical, Numerical, Graphical).

Increasing and Decreasing Functions

• Increasing Function: As input values increase (moving right), output values also increase.

  • Analytically: Given two values A and B where A < B, then F(A) < F(B).

• Decreasing Function: As input values increase, output values decrease.

  • Analytically: Given two values A and B where A < B, then F(A) > F(B).

Graphical Representation

• Zeros: Points where the graph intersects the X-axis (output = 0).

• Y-Intercept: Point where the graph intersects the Y-axis (input = 0).

Concavity of Graphs

• Concave Up: Shape resembles a bowl; slope is increasing.

• Concave Down: Shape resembles an upside-down bowl; slope is decreasing.

• Point of Inflection: The point where the curve changes from concave up to concave down or vice versa.

Identifying Features in Graphs

1. When is the graph concave up?

   • Written as inequalities or interval notation.

2. When is the graph concave down?

   • Similar writing method as above.

3. Finding Zeros: Identified by the output value being zero.

4. Finding Y-Intercepts: Identified by the input value being zero.

5. Increasing and Decreasing Intervals: Identified by analyzing the behavior of the graph.

Summary of Key Concepts

• Review variable representations, concavity, increasing/decreasing functions, zeros, and Y-intercepts.

• Practice writing interval notation and inequalities to express domain characteristics.

Tips for Success

1. Complete practice problems and check answers.

2. Collaborate with classmates and seek help from teachers if needed.

3. Focus on understanding core concepts and their applications.

Conclusion

• Emphasized main ideas in functions and their representations.

• Reminder to be kind and helpful in academic pursuits.