Chapter 2 – Linear and Exponential Functions

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

2. 2.1 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

3. 2.1 In summer of 2001, the exchange rate for the Mexican peso was 9.2. CONSTANT RATE OF CHANGE Mexican peso conversion is a linear function with respect to US dollar.

4. 2.1 In summer of 2001, the exchange rate for the Mexican peso was 9.2. pesos dollars straight line graph Mexican peso conversion is a linear function with respect to US dollar.

5. 2.1 In summer of 2001, the exchange rate for the Mexican peso was 9.2. p(d) = 0.92*d linear formula: f(x) = b + mx starting value/y-intercept (b) is 0. rate of change/slope (m) is 0.92. Mexican peso conversion is a linear function with respect to US dollar.

6. 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 . CONSTANT RATE OF CHANGE Value of DJ system is a linear function with respect to age.

7. 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 . value(dollars) age (years) straight line graph Value of DJ system is a linear function with respect to age.

8. 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 ($300) each year . v(t) = 3000 - 300*t linear formula: f(x) = b + mx starting value/y-intercept (b) is 3000 [$]. rate of change/slope (m) is -300 [$ per year]. Value of DJ system is a linear function with respect to age.

9. 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. CONSTANT RATE OF CHANGE Monthly bill is a linear function with respect to number of hours used.

10. 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. bill (dollars) time (hours) STRAIGHT LINE GRAPH Monthly bill is a linear function with respect to number of hours used.

11. 2.1 Under America Online’s Unlimited Usage plan, a member is charged $21.95 per month regardless of the number of hours spent online. Express the monthly bill as a function of the number of hours used in one month. U(t) = 21.95 linear formula: f(x) = b + mx starting value/y-intercept (b) is 21.95 [$]. rate of change/slope (m) is 0 [$ per hour]. Monthly bill is a linear function of number of hours spent online.

12. 2.1 Not all straight line graphs are linear functions. Consider the equation x = 3. linear formula: f(x) = b + mx

13. 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 * ax where f is the name of the function. c is the starting value or y intercept (f(0)). a is the growth factor.

14. 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. NO CONSTANT RATE OF CHANGE [increasing].

15. 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. Growth factor is 2 [doubling].

16. 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. bacteriapopulation time (20-minute intervals) GRAPH IS CONCAVE UP [increasing rate of change].

17. 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. P(t) = 2t exponential formula: f(x) = c*ax 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) = 224 = 16,777,216 bacteria

18. 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].

19. 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. “growth” factor is 0.75 [decreasing by 25% means 75% remains]

20. 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. number of polio cases years since 1988 GRAPH IS CONCAVE UP [increasing rate of change].

21. 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*ax starting value/y-intercept (c) is 38000 [polio cases].growth factor (a) is 0.75. Number of polio cases is an exponential function of time.

22. Chapter 2 – Linear and Exponential Functions HWp81: 1-6, 13-18, 21-23 TURN IN: #13, #16, #22,