Types of Functions, Rates of Change Lesson 1.4. Constant Functions Consider the table of ordered...
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Transcript of Types of Functions, Rates of Change Lesson 1.4. Constant Functions Consider the table of ordered...
Types of Functions, Rates of Change
Lesson 1.4
Constant Functions
Consider the table of ordered pairs
The dependent variable is the same It is constant
The graph is a horizontal line
Month 1 2 3 4 5 6
Rent Paid $735 $735 $735 $735 $735 $735
Rent Paid
$0$100$200$300$400$500$600$700$800
0 1 2 3 4 5 6 7( ) 735R m
Linear Function
Can be represented by Where a and b are constants
See Geogebra example
( )f x a x b
Slope and Y-Intercept
Considering y = m * x + b The b is the y-intercept
Where on the y-axis, the line intersects On your calculator
Go to Y= screen Enter at Y1 (2/3) * x + 5 Predict what the graph will look like before you
specify F2, 6 for standard zoom
Family of Linear Functions
Slope = Rate of Change
change in y
change in x
constant rate of change
y m x b
ym slope
x
y=3x + 5• Slope = m = 3
• y-intercept = b = 5
Slope and Y-Intercept
The function y = (2/3) * x + 5
Slope = 2/3 (up to the right) Y-intercept = 5
•
Linear Functions
Consider this set ofordered pairs
If we plot the pointsand join them wesee they lie in aline
x y
0 5
1 8
2 11
3 14
4 17
••
••
Rate of Change
Given function y = 3x + 5
••
••
x y
0 5
1 8
2 11
3 14
4 17
Change in y = y = 66
Change in x = = 2x
3
change in x 6Average rate of change = 3
change in y 2
y
y
Rate of Change
Try calculating for differentpairs of (x, y) points
You should discover that the rate of change is constant … in this case, 3
x y
0 5
1 8
2 11
3 14
4 17
1 2
1 2
change in xAverage rate of change =
change in y
y
y
y y
x x
GeogebraDemo
Slope
When slope =
Try y = -7x – 3(predict the results before you graph)
2 change in y
3 change in x
y
x
3x
2y
Family of Linear Functions
Calculating slope with two ordered pairs
•
•(X1, Y1)
(X2, Y2)
xy
1 2
1 2
y yslope
x x
Given two ordered pairs, (7,5) and (-3,12). What is the slope of the line
through these two points?
Given two ordered pairs, (7,5) and (-3,12). What is the slope of the line
through these two points?
5 12 70.7
7 ( 3) 10
Rate of Change
Consider the function Enter into Y= screen of calculator
View tables on calculator (♦ Y)
( )f x x
You may need to specify the
beginning x value and the increment
Rate of Change
As before, determine therate of change fordifferent sets of orderedpairs
x sqrt(x)
0 0.00
1 1.00
2 1.41
3 1.73
4 2.00
5 2.24
6 2.45
7 2.651 2
1 2
change in xAverage rate of change =
change in y
y
y
y y
x x
Rate of Change (NOT a constant)
You should find that the rate of change is changing – NOT a constant.
Contrast to thefirst functiony = 3x + 5 ( )f x x
GeogebraDemo
Function Defined by a Table
Consider the two functions defined by the table The independent variable is the year.
Predict whether or not the rate of change is constant
Determine the average rate of change for various pairs of (year, sales) values
Year 1982 1984 1986 1988 1990 1992 1994
CD sales 0 5.8 53 150 287 408 662
LP sales 244 205 125 72 12 2.3 1.9
Warning
Not all functions which appear linear will actually be linear!!
Consider the set of ordered pairs Graph them Decide whether graph
is linear Check slope for different
pairs
t P
0 67.38
1 69.13
2 70.93
3 72.77
4 74.67
5 76.61
6 78.60
Results
Graph appearsstraight
But …rate of change is not a constant
66
68
70
72
74
76
78
80
0 2 4 6 8
t P slope
0 67.38
1 69.13 1.75
2 70.93 1.8
3 72.77 1.84
4 74.67 1.9
5 76.61 1.94
6 78.6 1.99
Assignment
Lesson 1.4 Page 53 Exercises 1 – 65 EOO