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
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