Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 Chapter 1 Introduction to Functions and Graphs

2 Copyright © 2014, 2010, 2006 Pearson Education, Inc. 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 1.4

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 3 Linear Function A function f represented by f (x) = mx + b, where m and b are constants, is a linear function.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 4 Constant Function A function f represented by f (x) = b, where b is a constant (fixed number), is a constant function.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 5 Recognizing Linear Functions A car initially located 30 miles north of the Texas border, traveling north at 60 miles per hour is represented by the function f(x) = 60x + 30 and has the graph:

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 6 Rate of Change of a Linear Function In a linear function f, each time x increases by one unit, the value of f(x) always changes by an amount equal to m. That is, a linear function has a constant rate of change. The constant rate of change m is equal to the slope of the graph of f.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 7 Rate of Change of a Linear Function In our car example: Throughout the table, as x increases by 1 unit, y increases by 60 units. That is, the rate of change or the slope is 60.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 8 Slope of Line as a Rate of Change The slope m of the line passing through the points (x 1, y 1 ) and (x 2, y 2 ) is

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 9 Slope Positive Slope If the slope of a line is positive, the line rises from left to right. Slope 2 indicates that the line rises 2 units for every unit increase in x.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 10 Negative Slope If the slope of a line is negative, the line falls from left to right. Slope –1/2 indicates that the line falls 1/2 unit for every unit increase in x.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 11 Slope of 0 Slope 0 indicates that the line is horizontal.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 12 Slope is Undefined When x 1 = x 2, the line is vertical and the slope is undefined.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 13 Find the slope of the line passing through the points (  2, 3) and (1, –2). Plot these points together with the line. Interpret the slope. Solution Example:Calculating the slope of a line

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 14 Example:Calculating the slope of a line The slope –5/3 indicates that the line falls 5/3 units for each unit increase in x, or equivalently, the line falls 5 units for each 3-unit increase in x.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 15 Nonlinear Functions If a function is not linear, then it is called a nonlinear function. The graph of a nonlinear function is not a (straight) line. Nonlinear functions cannot be written in the form f(x) = mx + b.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 16 Graphs of Nonlinear Functions There are many nonlinear functions. Square FunctionSquare Root Function

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 17 Graphs of Nonlinear Functions Here are two other common nonlinear functions: Cube FunctionAbsolute Value Function

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 18 Increasing and Decreasing Functions Suppose that a function f is defined over an interval I on the number line. If x 1 and x 2 are in I, (a)f increases on I if, whenever x 1 < x 2, f(x 1 ) < f(x 2 ); (b)f decreases on I if, whenever x 1 f(x 2 ).

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 19 Graphs of Increasing and Decreasing Functions when x 1 < x 2, then f(x 1 ) < f(x 2 ) and f is increasing when x 1 f(x 2 ) and f is decreasing

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 20 Graphs of Increasing and Decreasing Functions If could walk from left to right along the graph of an increasing function, it would be uphill. For a decreasing function, we would walk downhill.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 21 Number Line Graph

Copyright © 2014, 2010, 2006 Pearson Education, Inc. Interval Notation A convenient notation for number line graphs is called interval notation. Instead of drawing number lines...

Copyright © 2014, 2010, 2006 Pearson Education, Inc. Closed and Open Intervals When a set includes the endpoints, the interval is a closed interval and brackets are used. When a set does not include the endpoints, the interval is an open interval and parentheses are used.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. Half-open Intervals When a set includes one endpoint and not the other, the interval is a half-open and 1 bracket and 1 parenthesis is used. This represents the interval

Copyright © 2014, 2010, 2006 Pearson Education, Inc. Union Symbol An inequality in the form x 3 indicates the set of real numbers that are either less than 1 or greater than 3. The union symbol U can be used to write this inequality in interval notation as

Copyright © 2014, 2010, 2006 Pearson Education, Inc. Interval Notation

Copyright © 2014, 2010, 2006 Pearson Education, Inc. Increasing, Decreasing, and Endpoints The concepts of increasing and decreasing apply only to intervals of the real number line and NOT to individual points. Do NOT say that the function f both increases and decreases at the point (0, 0). Decreasing: (–∞, 0] Increasing: [0, ∞)

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 28 Example: Determining where a function is increasing or decreasing Use the graph of Decreasing: and interval notation to identify where f is increasing or decreasing. Solution Increasing:

Copyright © 2014, 2010, 2006 Pearson Education, Inc. Average Rate of Change Graphs of nonlinear functions are not straight lines, so we speak of average rate of change. The line L is referred to as the secant line, and the slope of L represents the average rate of change from x 1 to x 2. Different values of x 1 and x 2 usually yield different secant lines and different average rates of change.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 30 Average Rate of Change Let (x 1, y 1 ) and (x 2, y 2 ) be distinct points on the graph of a function f. The average rate of change of f from x 1 to x 2 is That is, the average rate of change from x 1 to x 2 equals the slope of the line passing through (x 1, y 1 ) and (x 2, y 2 ).

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 31 Let f(x) = 2x 2. Find the average rate of change from x = 1 to x = 3. Calculate f(1) and f(3) The average rate of change is equals the slope of the line passing through the points (1, 2) and (3, 18). Example: Finding an average rate of change Solution

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 32 (1, 2) and (3, 18) The average rate of change from x= 1 to x = 3 is 8. Example: Finding an average rate of change

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 33 The Difference Quotient The difference quotient of a function f is an expression of the form where h ≠ 0.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 34 Let f(x) = 3x – 2. a. Find f(x + h) b. Find the difference quotient of f and simplify the result. (a) To find f(x + h), substitute (x + h) for x in the expression 3x – 2. Example: Finding a difference quotient Solution

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 35 (b) The difference quotient can be calculated as follows: Example: Finding a difference quotient