Section 1.3Linear Function

• Last section we discussed average rate of change over a certain interval

• When a function has a constant rate of change (i.e. the same average rate of change over all intervals) it’s graph is a line and we call it linear

Consider a car going 25 miles per hour and we get the following table

t 0 2 4 6 8 10

D 0 50 100 150 200 250

•What is the average rate of change for each interval?

•Find a formula for D as a function of t

•What form are linear functions in?

Where t is time in hours and D is distance in miles

Form of a linear function• For our example we had D = 25t

• Linear functions are most commonly written as y = b + mx or y = mx + b where m is the slope and b is the y-intercept where

and b gives the value for x = 0

• In general

output = rate of change x input + initial value

01

01

xxyy

xym

• Do the following problem in your groups• Suppose we know the population of a city is 23,000

in 1982 and 21,000 in 1986• Assuming the population has been declining at a

constant rate since 1970, find a formula for the population as a function of time, t– Let t be the number of years since 1970 (i.e. t = 0

corresponds to the year 1970)

• Use your model to determine the population in the year 2000

• When will the population be 0• Be prepared to present your solution to the class

• Do the following problem in your groups

• The profit, in dollars, of selling n items is given by P(n) = 0.98n – 3000– Identify the vertical intercept and the slope– Explain their meanings in practical terms

• Do the following problem in your groups

• A hotel finds that if it spends no money on renovations, they will be able to rent 100 rooms per night

• They find that for every $5000 spent on renovation, they will be able to rent an additional 25 rooms

• If x is the amount of money they spend and y is the number of rooms, find a formula for y as a function of x