Types of Functions, Rates of Change

Types of Functions, Rates of Change

Types of Functions, Rates of Change Lesson 1.4 Constant Functions Consider Month 1 2 3 4 5 6 Rent Paid $735 $735 $735 $735 $735 $735

The the table of ordered pairs dependent variable is the same It is constant The graph is a horizontal line R (m) 735 Rent Paid $800 $700 $600 $500 $400 $300 $200 $100 $0 0 1 2 3

4 5 6 7 Linear Functions Consider this set of ordered pairs If we plot the points and join them we see they lie in a line x y 0 5 1

8 2 11 3 14 4 17 Rate of Change Given function y = 3x + 5

Change in y = y = 6 Change in x = x = 2 Average rate of change = 3 x y 0 5 1 8 2 11 3 14 4 17

change in x y 6 3 change in y y 2 6 Rate of Change Try calculating for different pairs of (x, y) points change in x y Average rate of change = change in y y y1 y2 x1 x2 x y 0 5 1 8 2

11 3 14 4 17 You should discover that the rate of change is constant in this case, 3 Family of Linear Functions Slope = Rate of Change y m x b y change in y m slope x change in x constant rate of change y=3x + 5 Slope = m = 3 y-intercept = b = 5 Slope and Y-Intercept Considering

y=m*x+b The b is the y-intercept Where On on the y-axis, the line intersects 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 Slope and Y-Intercept The function y = (2/3) * x + 5 Slope = 2/3 (up to the right) Y-intercept = 5 Slope When slope =

2 change in y y 3 change in x x y 2 x 3 Try y = -7x 3 (predict the results before you graph) Family of Linear Functions Calculating slope with two ordered pairs (X2, Y2) (X1, Y1) y x Given Giventwo twoordered orderedpairs,

pairs,(7,5) (7,5)and and (-3,12). (-3,12). What Whatisisthe theslope slopeofofthe theline line through throughthese thesetwo twopoints? points? y1 y2 slope x1 x2 5 12 7 0.7 7 ( 3) 10 Rate of Change the function f ( x ) x Enter into Y= screen of calculator Consider View tables on calculator ( Y)

You may need to specify the beginning x value and the increment Rate of Change As before, determine the rate of change for different sets of ordered pairs change in x y Average rate of change = change in y y y1 y2 x1 x2 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 2.65 Rate of Change (NOT a constant) You should find that the rate of change is changing NOT a constant. Contrast to the first function y = 3x + 5 f ( x) x

Function Defined by a Table 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 Consider The the two functions defined by the table 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 Warning Not all functions which appear linear will actually be linear!! t P Consider the set of 0 67.38 ordered pairs 1 69.13 Graph them Decide whether graph

is linear Check slope for different pairs 2 3 4 5 6 70.93 72.77 74.67 76.61 78.60 Results 80 78 76 Graph appears straight But rate of change is not a constant 74 72 70

68 66 0 2 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 4 6 8 Assignment Lesson 1.4 Page 53 Exercises 1 71 EOO

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