Chapter 1 – Introducing Functions

Section 1.1 – Defining Functions

Definition of a FunctionFormulas, Tables and GraphsFunction Notation

Section 1.2 – Using Functions to Model the Real World

Abstract and ModelDomain and RangeDiscrete and ContinuousOne Output for Each Input

Exercises from Sections 1.1 and 1.2

1/10 – domain of men’s shoe function

5/10 – domain

6/10 - domain

2/13 – function notation

3/13 – function notation

4/13 – function notation

5/13 – function notation

6/13 – function notation

11/32 – function notation

12/32 – function notation

23/33 – weird function

Section 1.3Watching Functions Values Change

pages 17 - 29

Section 1.3 – Watching Function Values ChangeExample 1: women’s shoe size function

w(x) = 3x – 21 y = 3x - 21

y (shoe size) depends on x (foot measurement)HOW?

foot length increases from 9 inches to 9 1/6 inches (by 1/6)

foot length increases from 9 1/6 to 9 1/3 inches (by 1/6)

foot length increase from 10 inches to 10 ½ inches (by 1/2)

foot length increases by a full inch during the course of a career.

What is the change in shoe size when:

x(inches)

9 9 1/6 9 1/3 10 10 1/2

y=3x-21(shoe size)

6 6.5 7 9 10.5

foot length increases from 9 inches to 9 1/6 inches (by 1/6)

foot length increases from 9 1/6 to 9 1/3 inches (by 1/6)

foot length increase from 10 inches to 10 ½ inches (by 1/2)

foot length increases by a full inch during the course of her career.

shoe size increases by 0.5

shoe size increases by 0.5

shoe size increases by 1.5

shoe size increases by 3

Example 1

Do you see a pattern?

Change in x(inches)

1/6 1/6 1/2 1

Change in y(shoe size)

0.5 0.5 1.5 3

change in y is always 3 times the change in x

or

change of one unit in x always produces a change of 3 units in y

or

rate of change is 3 [shoe sizes per 1 inch]

women’s shoe size functionw(x) = 3x – 21 y = 3x - 21

y (shoe size) depends on x (foot measurement)HOW?

Example 1

Section 1.3 – Watching Function Values Change

Definition (pg 17)The rate of change of y with respect to x is given by:

change in y-value

change in x value

Definition (pg 18)The average rate of change of any function y = f(x) from x = a to x = b is the ratio:

f(b) - f(a)

b - a

women’s shoe size functionw(x) = 3x – 21 y = 3x - 21

y (shoe size) depends on x (foot measurement)HOW?

Use the definition to determine the average rate of change of the women’s shoe size function for:

x from 9 to 9 1/6

x from 9 1/6 to 9 1/3

x from 10 to 10 1/2

x from 9 to 10

x from a to b

Example 1

Can we see the rate of change on the graph of the

women’s shoe size function?

Constant Rate of Change = Straight Line Graph

Example 1

Section 1.3 – Watching Function Values Change

Example 2the area of a square depends on the length of its side

A(s) = s2

HOW?

Use the definition to determine the average rate of change of the area function for:

s from 1 to 2

s from 2 to 3

s from 3 to 4

Use the figures to determine the average rate of change of the area function for:

s from 1 to 2

s from 2 to 3

s from 3 to 4

Example 2

Can we see the rates of change on the graph of the

area function?

Example 2

Rate of Change is Changing

Section 1.3 – Watching Function Values Change

The Shape of a Graph

We say a function is increasing if the value of the dependent variable increases as the value of the

independent variable increases

i.e. as we read the graph from left to right, the y values of points on the graph get larger.

i.e. the graph rises as we read left to right

Section 1.3 – Watching Function Values Change

The Shape of a Graph

We say a function is decreasing if the value of the dependent variable decreases as the value of the

independent variable increases

i.e. as we read the graph from left to right, the y values of points on the graph get smaller.

i.e. the graph falls as we read left to right

Section 1.3 – Watching Function Values Change

The Shape of a Graph

What about average rates of change for increasing functions?

2( ) 5f x x x ( ) 3f x x

Section 1.3 – Watching Function Values Change

The Shape of a Graph

What about average rates of change for decreasing functions?

2( ) 5f x x x ( ) 3f x x

Section 1.3 – Watching Function Values Change

The Shape of a Graph

We say the graph of a function is concave up if the rates of change increase as we move left to right.

We say the graph of a function is concave down if the rates of change decrease as we move left to right.

Where is f increasing?

Where is f decreasing?

Where is graph concave up?

Where is graph concave down?

Homework:

Page 33: #25-31Page 40: #1, 2

Turn in: 26,27,30, 31

Read and begin work on Lab 1B (pp 46-50)

Section 1.3 – Watching Function Values Change