Chapter 2 – Linear and Exponential Functions 2.1 – Introducing Linear Models 2.2 – Introducing Exponential Models 2.3 – Linear Model Upgrades

A linear function models any process that has a constant rate of change. m = The graph of a linear function is a straight line. A linear function has the form: y = f(x) = b + mx where f is the name of the function. b is the starting value or y intercept (f(0)). m is the constant rate of change or slope. slope intercept form 2.1

In summer of 2001, the exchange rate for the Mexican peso was 9.2. x (dollar) y (peso) x 0 to 1 1 to 2 0 to 3 1 to 10 change in x 1139 y 0 to to to to 92 change in y m9.2/ / / /99.2 Mexican peso conversion is a linear function with respect to US dollar. CONSTANT RATE OF CHANGE 2.1

dollars pesos straight line graph Mexican peso conversion is a linear function with respect to US dollar. In summer of 2001, the exchange rate for the Mexican peso was

p(d) = 0.92*d linear formula: f(x) = b + mx starting value/y-intercept (b) is 0. rate of change/slope (m) is Mexican peso conversion is a linear function with respect to US dollar. In summer of 2001, the exchange rate for the Mexican peso was

Jason decides to purchase a $3000 DJ system that has a life expectancy of 10 years. He assumes the value of the equipment will depreciate linearly by the same amount ($300) each year. x (age) y (value) x 0 to 1 1 to 2 0 to 5 3 to 10 change in x 1157 y 3000 to to to to 0 change in y m-300/ / / /7-300 CONSTANT RATE OF CHANGE Value of DJ system is a linear function with respect to age. 2.1

Jason decides to purchase a $3000 DJ system that has a life expectancy of 10 years. He assumes the value of the equipment will depreciate linearly by the same amount ($300) each year. straight line graph Value of DJ system is a linear function with respect to age. 2.1 age (years) value (dollars)

v(t) = *t linear formula: f(x) = b + mx starting value/y-intercept (b) is 3000 [$]. rate of change/slope (m) is -300 [$ per year]. Jason decides to purchase a $3000 DJ system that has a life expectancy of 10 years. He assumes the value of the equipment will depreciate linearly by the same amount ($300) each year. Value of DJ system is a linear function with respect to age. 2.1

Under America Online’s Unlimited Usage plan, a member is charged $21.95 per month regardless of the number of hours spent online. Express the monthly bill as a function of the number of hours used in one month. t (hours) bill (dollars) x 0 to 1 1 to 2 2 to 10 1 to 20 change in x y to change in y 0000 m0/100/100/800/190 CONSTANT RATE OF CHANGE Monthly bill is a linear function with respect to number of hours used. 2.1

Under America Online’s Unlimited Usage plan, a member is charged $21.95 per month regardless of the number of hours spent online. Express the monthly bill as a function of the number of hours used in one month. STRAIGHT LINE GRAPH Monthly bill is a linear function with respect to number of hours used. 2.1 time (hours) bill (dollars)

U(t) = linear formula: f(x) = b + mx starting value/y-intercept (b) is [$]. rate of change/slope (m) is 0 [$ per hour]. Monthly bill is a linear function of number of hours spent online. Under America Online’s Unlimited Usage plan, a member is charged $21.95 per month regardless of the number of hours spent online. Express the monthly bill as a function of the number of hours used in one month. 2.1

Not all straight line graphs are linear functions. Consider the equation x = 3.x33333y-4035 x 3 to 3 change in x 0000 y -4 to 1 -4 to 0 -1 to 0 0 to 5 change in y 5415 m5/0u4/0u1/0u5/0u linear formula: f(x) = b + mx 2.1

An exponential function models any process in which function values change by a fixed ratio or percentage. The graph of an exponential function is curvy. An exponential function has the form: y = f(x) = c * a x where f is the name of the function. c is the starting value or y intercept (f(0)). a is the growth factor.

2.2 Harmful kitchen bacteria can double their numbers every 20 minutes. A single bacterium on a wet countertop might in just eight hours, reproduce to nearly 17 million. t (20 minute intervals) P (number of bacteria) t 0 to 1 1 to 2 change in t 11 P 1 to 2 2 to 4 change in P 12 m1/112/12 NO CONSTANT RATE OF CHANGE [increasing].

2.2 Harmful kitchen bacteria can double their numbers every 20 minutes. A single bacterium on a wet countertop might in just eight hours, reproduce to nearly 17 million. t (20 minute intervals) P (number of bacteria) Growth factor is 2 [doubling]. ratio of consecutive output values tP(t+1)/P(t) 0 P(1)/P(0) = 2 / 1 = 2 1 P(2)/P(1) = 4 / 2 = 2 2 P(3)/P(2) = 8 / 4 = 2

GRAPH IS CONCAVE UP [increasing rate of change]. Harmful kitchen bacteria can double their numbers every 20 minutes. A single bacterium on a wet countertop might in just eight hours, reproduce to nearly 17 million. time (20-minute intervals) bacteria population

P(t) = 2 t exponential formula: f(x) = c*a x starting value/y-intercept (c) is 1 [bacteria]. growth factor (a) is 2. Bacteria population is an exponential function of time. After 8 hours (24 20-minute time intervals): P(24) = 2 24 = 16,777,216 bacteria Harmful kitchen bacteria can double their numbers every 20 minutes. A single bacterium on a wet countertop might in just eight hours, reproduce to nearly 17 million.

t (years since 1988) 0123 P (polio cases) 38, *38000 = *28500 = *21375 = t 0 to 1 1 to 2 change in t 11 P to to change in P m During the late twentieth century, WHO adopted as one of its goals the elimination of polio throughout the world. From 1988 to 1996, cases of polio decreased by roughly 25% annually. NO CONSTANT RATE OF CHANGE [increasing].

t (years since 1988) 0123 P (polio cases) 38, *38000 = *28500 = *21375 = During the late twentieth century, WHO adopted as one of its goals the elimination of polio throughout the world. From 1988 to 1996, cases of polio decreased by roughly 25% annually. ratio of consecutive output values tP(t+1)/P(t) 0 P(1)/P(0) = / =.75 1 P(2)/P(1) = / =.75 2 P(3)/P(2) = / =.7499 “growth” factor is 0.75 [decreasing by 25% means 75% remains]

During the late twentieth century, WHO adopted as one of its goals the elimination of polio throughout the world. From 1988 to 1996, cases of polio decreased by roughly 25% annually. GRAPH IS CONCAVE UP [increasing rate of change]. years since 1988 number of polio cases

During the late twentieth century, WHO adopted as one of its goals the elimination of polio throughout the world. From 1988 to 1996, cases of polio decreased by roughly 25% annually. P(t) = 38000*(.75) t exponential formula: f(x) = c*a x starting value/y-intercept (c) is [polio cases]. growth factor (a) is Number of polio cases is an exponential function of time.

Chapter 2 – Linear and Exponential Functions HWp81: 1-6, 13-18, TURN IN: #13, #16, #22,