Module3

Module Overview

  • NC State University Module 3: Working with Functions

Objectives

  1. Draw graphs of basic "toolkit" functions and their domain and range.

  2. Calculate the average rate of change over different intervals with different functions.

  3. Graph functions using horizontal and vertical transformations.

  4. Identify and apply shifts, reflections, and stretches to the graphs of functions.

Average Rate of Change

Definition

  • A rate of change describes how an output quantity changes relative to a change in the input quantity.

  • Units: "output units per input units."

  • Formula for the average rate of change between two input values:[ \text{Average Rate of Change} = \frac{\Delta y}{\Delta x} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} ]

Example Calculations

  1. Example 1: Find the average rate of change of f(x) = x² on [1, 2].

    • f(1) = 1² = 1

    • f(2) = 2² = 4

    • Average rate of change = (4 - 1) / (2 - 1) = 3

  2. Example 2: For f(x) = x over the interval [1, 1 + h].

    • Calculation leads to average rate of change as h.

  3. Example 3: For f(x) = 1/x over the interval [1, 1 + h].

    • Evaluate to find the average rate of change in terms of h.

  4. Example 4: For f(x) = 2x² + 1 on [x, x + h].

    • Show calculations leading to the average rate of change expression.

Real-World Example

  1. Speed Calculation:

    • Anna records her distance from home over time.

    • t (hours): [0, 1, 2, 3, 4, 5, 6, 7]

    • D(t) (miles): [10, 55, 90, 153, 214, 240, 292, 300]

    • Average speed over the first 6 hours = (D(6) - D(0)) / (6 - 0) = (292 - 10) / 6 = 46 miles/hour.

  2. Odometer Example:

    • Initial reading: 21395 miles

    • Final reading after 13.5 hours: 22125 miles

    • Average speed = (22125 - 21395) / 13.5 = 54.1 miles/hour.

Increasing and Decreasing Functions

Definitions

  • Increasing Function: Values increase as input increases.

  • Decreasing Function: Values decrease as input increases.

  • Average rate of change:

    • Positive for increasing functions.

    • Negative for decreasing functions.

Function Intervals

  1. Increasing on the intervals: (-∞, -2) ∪ (2, ∞)

  2. Decreasing on the interval: (-2, 2)

Local Maxima and Minima

Local Maximum

  • A local maximum occurs where the graph of a function is higher than at nearby points on both sides.

Local Minimum

  • A local minimum occurs where the graph is lower at nearby points on both sides.

Examples of Functions

  1. Function f(x):

    • Local maximum at (-2, 16)

    • Local minimum at (2, -16)

  2. Function f(x) with intervals identified:

    • Increasing on (1, 3) ∪ (4, ∞)

    • Decreasing on (-∞, 1) ∪ (3, 4)

    • Local minimum at (1, -1) and local maximum at (3, 1).

Toolkit Functions

Types of Functions

  1. Constant Function: Neither increasing nor decreasing.

  2. Identity Function: Increasing on (0, ∞).

  3. Quadratic Function: Decreasing on (-∞, 0), minimum at x=0.

  4. Cubic Function: Increasing.

  5. Reciprocal Function: Decreasing on (-∞, 0) and (0, ∞).

  6. Square Root Function: Increasing on (0, ∞).

  7. Absolute Value Function: Increasing on (0, ∞), decreasing on (-∞, 0).

Transformations of Functions

Types of Transformations

  1. Vertical and Horizontal Shifts:

    • Right h units: y = f(x - h)

    • Left h units: y = f(x + h)

    • Up k units: y = f(x) + k

    • Down k units: y = f(x) - k

  2. Reflections:

    • Vertical: y = -f(x) (flips over x-axis)

    • Horizontal: y = f(-x) (flips over y-axis)

  3. Stretches and Compressions:

    • Vertical: G(x) = af(x) (a > 1 stretches, 0 < a < 1 compresses)

    • Horizontal: G(x) = f(bx) (b > 1 compresses, 0 < b < 1 stretches)

Example of Transformations

  1. Graphing example: y = 2x - 3 + 4.

    • Analyze shifts, stretching, and domain/range.

  2. Additional graphing examples with increasing/decreasing intervals and domain/range identification.