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Section 1.3 Linear Functions 2

Constant Rate of Change In the previous section, we introduced the average rate of change of a function on an interval. For many functions, the average rate of change is different on different intervals. For the remainder of this chapter, we consider functions which have the same average rate of change on every interval. Such a function has a graph which is a line and is called linear. Page 173

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (a) What is the average rate of change of P over every time interval? (b) Make a table that gives the town's population every five years over a 20-year period. Graph the population. (c) Find a formula for P as a function of t. Page 18 (Example 1)4

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (a) What is the average rate of change of P over every time interval? This is given in the problem: 2,000 people / year Page 185

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (b) Make a table that gives the town's population every five years over a 20-year period. Graph the population. Page 186

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (b) Make a table that gives the town's population every five years over a 20-year period. Graph the population. t, yearsP, population Page 187

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (b) Make a table that gives the town's population every five years over a 20-year period. Graph the population. t, yearsP, population 030, , , , ,000 Page 18 8

(b) Make a table that gives the town's population every five years over a 20-year period. Graph the population. Page 189

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (c) Find a formula for P as a function of t. Page 18 10

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (c) Find a formula for P as a function of t. We want: P = f(t) Page 18 11

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (c) Find a formula for P as a function of t. We want: P = f(t) If we define: P = initial pop + (growth/year)(# of yrs) Page 18 12

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (c) Find a formula for P as a function of t. tP 030, , , , ,000 If we define: P = initial pop + (growth/year)(# of yrs) Page 18 13

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (c) Find a formula for P as a function of t. tP 030, , , , ,000 We substitute the initial value of P: P = 30,000 + (growth/year)(# of yrs) Page 18 14

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (c) Find a formula for P as a function of t. tP 030, , , , ,000 And our rate of change: P = 30,000 + (2,000/year)(# of yrs) Page 18 15

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (c) Find a formula for P as a function of t. tP 030, , , , ,000 And we substitute in t: P = 30,000 + (2,000/year)(t) Page 18 16

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (c) Find a formula for P as a function of t. tP 030, , , , ,000 Our final answer: P = 30, ,000t Page 18 17

Here again is the graph and the function. Page 18 18

Any linear function has the same average rate of change over every interval. Thus, we talk about the rate of change of a linear function. In general: A linear function has a constant rate of change. The graph of any linear function is a straight line. Page 19 19

Depreciation Problem A small business spends $20,000 on new computer equipment and, for tax purposes, chooses to depreciate it to $0 at a constant rate over a five-year period. (a) Make a table and a graph showing the value of the equipment over the five-year period. (b) Give a formula for value as a function of time. Page 19 (Example 2) 20

Used by economists/accounts: a linear function for straight-line depreciation. Example: tax purposes-computer equipment depreciates (decreases in value) over time. Straight-line depreciation assumes: the rate of change of value with respect to time is constant. Page 19 21

t, yearsV, value ($) Let's fill in the table: Page 19 22

t, yearsV, value ($) Let's fill in the table: Page 1923

t, yearsV, value ($) 0$20,000 1$16,000 2$12,000 3$8,000 4$4,000 5$0 Let's fill in the table: Page 19 24

And our graph: Page 19 25

Give a formula for value as a function of time: Page 19 26

Give a formula for value as a function of time: Page 19 27

Give a formula for value as a function of time: Page 19 28

Give a formula for value as a function of time: Page 19 29

Give a formula for value as a function of time: Page 19 30

Give a formula for value as a function of time: More generally, after t years? Page 19 31

Give a formula for value as a function of time: More generally, after t years? Page 19 32

Give a formula for value as a function of time: What about the initial value of the equipment? Page 19 33

Give a formula for value as a function of time: What about the initial value of the equipment? $20,000 Page 19 34

Give a formula for value as a function of time: What about the initial value of the equipment? $20,000 What is our final answer for the function? Page 19 35

Give a formula for value as a function of time: What about the initial value of the equipment? $20,000 What is our final answer for the function? V = 20, ,000t Page 19 36

Let's summarize: y x m b Page 20 37

Let's summarize: y x m b b = y intercept (when x=0) m = slope Page 20 38

Let's summarize: y x m b y = b + mx Page 20 39

Let's summarize: y x m b Page 20 40

Let's summarize: y x m b Page 20 41

Let's recap: example #1: P = 30, ,000t m = ? b = ? Page 20 42

Let's recap: example #1: P = 30, ,000t m = 2,000 b = 30,000 Page 20 43

Let's recap: example #2: V = 20, ,000t m = ? b = ? Page 20 44

Let's recap: example #2: V = 20, ,000t m = -4,000 b = 20,000 Page 20 45

Can a table of values represent a linear function? Page 21 46

Could a table of values represent a linear function? Yes, it could if: Page 21 47

Could a table of values represent a linear function? Yes, it could if: Page 21 48

xp(x)ΔxΔxΔpΔpΔp/Δx Could p(x) be a linear function? Page 2149

xp(x)ΔxΔxΔpΔpΔp/Δx Could p(x) be a linear function? Page 2150

xp(x)ΔxΔxΔpΔpΔp/Δx Could p(x) be a linear function? Page 2151

xp(x)ΔxΔxΔpΔpΔp/Δx Could p(x) be a linear function? Page 2152

xp(x)ΔxΔxΔpΔpΔp/Δx Since Δp/Δx is constant, p(x) could represent a linear function. Page 2153

xq(x)ΔxΔxΔqΔqΔq/Δx Could q(x) be a linear function? Page 2154

xq(x)ΔxΔxΔqΔqΔq/Δx Could q(x) be a linear function? Page 2155

xq(x)ΔxΔxΔqΔqΔq/Δx Could q(x) be a linear function? Page 2156

xq(x)ΔxΔxΔqΔqΔq/Δx Could q(x) be a linear function? Page 2157

xq(x)ΔxΔxΔqΔqΔq/Δx Since Δq/Δx is NOT constant, q(x) does not represent a linear function. Page 2158

Yearp, Price ($) Q, # sold (cars) ΔpΔpΔQΔQΔQ/Δp 19853,99049, ,11043, ,20038, ,33032,000 What about the following example? Yugos exported from Yugoslavia to US. Page 22 59

Yearp, Price ($) Q, # sold (cars) ΔpΔpΔQΔQΔQ/Δp 19853,99049, ,11043, ,20038, ,33032,000 What about the following example? Yugos exported from Yugoslavia to US. Page 22 60

Yearp, Price ($) Q, # sold (cars) ΔpΔpΔQΔQΔQ/Δp 19853,99049, , ,11043, , ,20038, , ,33032,000 What about the following example? Yugos exported from Yugoslavia to US. Page 22 61

Yearp, Price ($) Q, # sold (cars) ΔpΔpΔQΔQΔQ/Δp 19853,99049, , cars/$ 19864,11043, , cars/$ 19874,20038, , cars/$ 19884,33032,000 What about the following example? Yugos exported from Yugoslavia to US. Page 2262

ΔpΔpΔQΔQΔQ/Δp 120-6, cars/$ 90-4, cars/$ 130-6, cars/$ Although Δp and ΔQ are not constant, ΔQ/Δp is. Therefore, since the rate of change (ΔQ/Δp) is constant, we could have a linear function here. Page 2263

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The function P = 100(1.02) t approximates the population of Mexico in the early 2000's. Here P is the population (in millions) and t is the number of years since Table 1.25 and Figure 1.21 show values of P over a 5-year period. Is P a linear function of t? Page 2365

t, yearsP (mill.)ΔtΔtΔPΔPΔP/Δt Page 23 66

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t, yearsP (mill.)ΔtΔtΔPΔPΔP/Δt Page 2468

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The formula P = 100(1.02) t is not of the form P = b + mt, so P is not a linear function of t. Page 2470

This completes Section 1.3.