Types of Functions and Their Rates of Change Chapter 1.4 Types of Functions and Their Rates of Change

Identify linear functions Interpret slope as a rate of change Identify nonlinear functions Identify where a function is increasing or decreasing Use interval notation Use and interpret average rate of change Calculate the difference quotient

Teaching Example 1 Solution Find the slope of the line passing through (−1, 2) and (3, −4). Explain what the slope indicates about the line. Solution The line falls 3/2 units for each unit increase in x.

Teaching Example 2 Solution The number of gallons of water remaining in a 100-gallon tank after x-minutes is given by G(x) = 100 – 5x. a. Evaluate G(5) and interpret the result. b. Find the slope of the graph of G. Interpret this slope as a rate of change. Solution a. G(5) = 100 – 5(5) = 100 – 25 = 75 After 5 minutes the tank contains 75 gallons of water. 4

Teaching Example 2 (cont) The number of gallons of water remaining in a 100-gallon tank after x-minutes is given by G(x) = 100 – 5x. a. Evaluate G(5) and interpret the result. b. Find the slope of the graph of G. Interpret this slope as a rate of change. Solution b. m = –5. Water is leaving the tank at a rate of 5 gal/min. 5

Teaching Example 3 Solution In 1985, $95 billion was spent on advertising in the United States, and in 2005, $270 billion was spent. Find and interpret the slope of the line passing through (1985, 95) and (2005, 270). Solution This means that advertising spending increased, on average, by $8.75 billion per year from 1985 to 2005. 6

Teaching Example 4 Solution Sketch a graph of y = x – 1. Determine the slope; y-intercept, x-intercept, formula for f(x) and any zeros. Solution slope = y-intercept = –1 x-intercept = 1 f(x) = x – 1 The zero of the function is 1. 7

Teaching Example 5 Solution For each function f, determine where f is increasing, and where f is decreasing. a. f(x) = x2 b. f(x) = 1 – 2x c. f(x) = |x – 2| Solution a. Increasing when x > 0 Decreasing when x < 0 8

Teaching Example 5 (cont) For each function f, determine where f is increasing, and where f is decreasing. a. f(x) = x2 b. f(x) = 1 – 2x c. f(x) = |x – 2| Solution b. never increasing Decreasing for all real numbers 9

Teaching Example 5 (cont) For each function f, determine where f is increasing, and where f is decreasing. a. f(x) = x2 b. f(x) = 1 – 2x c. f(x) = |x – 2| Solution c. Increasing when x > 2 Decreasing when x < 2 10

Teaching Example 6 Solution Determine where is increasing or decreasing. Solution f(x) is increasing on and decreasing on 11

Teaching Example 7 Solution Let f(x) = x2. Find the average rate of change from x = 2 to x = 4. Solution 12

Teaching Example 8 Solution If f(1) = 12 and f(4) = −3, find the average rate of change of f from 1 to 4. Solution The average rate of change of f from 1 to 4 is −5.

Teaching Example 9 Solution Find the difference quotient for f(x) = –2x + 5 Solution

Teaching Example 10 Find the difference quotient for Solution 15