Review #2: Linear and exponential functions

Transcription

1 June 12, 2009 Review #2: Linear and exponential functions page 1 Review #2: Linear and exponential functions Comparison: linear function vs. exponential function Linear Function y = mx + b or y = m(x x 1 ) + y 1 NEXT = NOW + m starting from b b is the starting value, m is the rate or the slope. m is positive for growth, negative for decay. Exponential Function y = a b x NEXT = NOW b starting from a a is the starting value, b is the base or the multiplier. b > 1 for growth, 0 < b < 1 for decay. See below for ways to find the base b. Growth and decay problems: how to decide linear vs. exponential If the growth or decay involves increasing or decreasing by a fixed number, use a linear function. The equation will look like: f(x) = (rate) x + (starting amount). If the growth or decay is expressed using a percent, a fraction, or multiplication (including words like doubling or tripling ) use an exponential function. The equation will look like: f(x) = (starting amount) (base) x. There are several ways to find the base, depending on the information given in the problem. o If you re given a growth rate as a percent: base = 1 + (growth rate as a decimal). o If you re given a decay rate as a percent: base = 1 (decay rate as a decimal). o If you re given a growth rate as a fraction: base = 1 + fraction. o If you re given a decay rate as a fraction: base = 1 fraction. o If you re given a number that s used for repeated multiplication, base = that number. (For example, if an amount triples every day, then base = 3.) o If you re given an input-output table or a graph: Use ExpReg on your calculator. Examples a. The current population of Smallville is 500 people. Suppose that the population decreases by 40 people each year. Let f(x) stand for what the population will be after x years. Write a function formula for f(x). Choice: The decrease is by a fixed number (40 people) so choose a linear function. Answer: f(x) = 40x b. The current population of Tinytown is 500 people. Suppose that the population decreases by 10% each year. Let g(x) stand for what the population will be after x years. Write a function formula for g(x). Choice: The decrease is by a percent rate (10%) so choose an exponential function. base = 1 (decay rate as decimal) = 1 (0.10) = Answer: f(x) = 500 (0.90) x

2 June 12, 2009 Review #2: Linear and exponential functions page 2 Linear functions: different ways to find the y= or f(x)= equation The equations below all use the y = form but any of them could be written f(x) = also. If you know the slope and the y-intercept y = mx + b where m is the slope, b is the y-intercept If you know the slope and a point y = m(x x 1 ) + y 1 where m is the slope and (x 1, y 1 ) is the point that you know If you know the slope and a point (ALTERNATE METHOD) Write y = mx + b. Put in the given slope as m, and put in the given point as x and y. Solve for b. Write y = mx + b again. Put in the given slope for m and the number you found for b. If you know the coordinates of two points First calculate the slope of the line using m = y 2 " y 1 x 2 " x 1. Then pick just one of the two points, and use the y = m(x x 1 ) + y 1 equation. In a word problem where you have a rate and initial value m = the number in the problem that is a rate or a speed b = the number in the problem that is an initial value or starting value Then use the equation y = mx + b, and put in the numbers for m and b. In a word problem where you have a rate, along with a pair of numbers m = the number in the problem that is a rate or a speed The pair of numbers give you a known point (x, y) See instructions above for If you know the slope and a point. In a word problem where you have two pairs of numbers The pairs of numbers give you two known points (x 1, y 1 ) and (x 2, y 2 ) Find the slope using m = y 2 " y 1 x 2 " x 1. Then pick just one of the two points, and use the y = m(x x 1 ) + y 1 equation.

3 June 12, 2009 Review #2: Linear and exponential functions page 3 Problems 1. a. The tuition at Franklin University is currently $20,000 per year. It is increasing by $1,000 each year. Let f(x) stand for what the tuition will be x years from now. Write a function formula for f(x). b. The tuition at Garfield College is currently $18,000 per year, and is increasing by 7% each year. Let g(x) stand for what the tuition will be x years from now. Write a function formula for g(x). c. Which school will have the smaller tuition, 4 years from now? Show your work. 2. Each table below shows either a linear function or an exponential function. Write a y = equation and a NOW-NEXT description for each. a. x y y =. NEXT =, starting from. b. x y y =. NEXT =, starting from.

4 June 12, 2009 Review #2: Linear and exponential functions page 4 3. A cell phone plan charges a fixed monthly fee, plus $0.10 per minute for calls. Someone who makes 400 minutes of calls in a month has to pay $ Let x stand for the number of minutes, and let f(x) stand for the total monthly charge. a. Find a function formula for f(x). b. What would be the charge for someone who makes 600 minutes of calls? c. If someone s cell phone charge is $75.00, how many minutes of calls did they make? d. What is the amount of the fixed monthly fee?

5 June 12, 2009 Review #2: Linear and exponential functions page 5 4. A classroom has a supply of 200 pencils at the start of the school year. Each month, 8 1 of the pencil supply is used up or lost. a. Complete this table showing the number of pencils after each month. month number number of pencils b. Write a function formula showing the number of pencils after x months. c. The teacher wants to order more pencils when there are 20 pencils remaining. After how many months will this happen? 5. a. Which of these equations is an exponential decay function? y = 0.07x + 8 y = 0.07x 8 y = 8 (1.07) x y = 8 (0.97) x. b. For the equation you picked, what is the starting value? c. For the equation you picked, what is the percent rate of decrease?

6 June 12, 2009 Review #2: Linear and exponential functions page 6 6. Sketch the typical graph shape for each type of function. The first one is done for you. linear growth linear decay exponential growth exponential decay 7. Here is a table showing the value of a computer as a function of its age. Let V(t) stand for the value of the computer after t years. t (age in years) V(t) (value in dollars) 0 $ $ $ $900 4 $700 a. Which should be used for this function: a linear function or an exponential function? Explain the reason for your choice. b. Write a function formula for V(t). c. According to this function, after how much time will the computer be worthless?

7 June 12, 2009 Review #2: Linear and exponential functions page 7 8. Here is a table showing the price of a car as a function of its age. Let P(t) stand for the price of the car after t years. t (age in years) P(t) (price in dollars) 0 $20,000 1 $18,000 2 $16,200 3 $14,580 4 $13,122 a. Which should be used for this function: a linear function or an exponential function? Explain the reason for your choice. b. What is the percent rate of decrease each year? c. Write a function formula for P(t). d. After how many years will the car be worth $5,000? 9. At the start of a lab experiment, there were 500 bacteria in a dish. Each hour, the number of bacteria doubled. a. After 3 hours, how many bacteria were there in the dish? b. After x hours, how many bacteria were there in the dish? c. The experiment was stopped when the number of bacteria in the dish reached 1 million. About how many hours long did the experiment take?

8 June 12, 2009 Review #2: Linear and exponential functions page Here are some facts about the Fahrenheit and Celsius temperature scales. Water freezes at 32º Fahrenheit or 0º Celsius. Water boils at 212º Fahrenheit or 100º Celsius. If you let F stand for Fahrenheit temperature and C stand for Celsius temperature, the relationship between F and C is a linear function. a. Find the linear equation that relates F and C. b. A typical human body temperature is 98.6º Fahrenheit. What is this temperature in Celsius degrees? c. An air conditioner turns on whenever the Celsius temperature is more than 25º. What is equivalent Fahrenheit temperature? d. There is a number of degrees where the Fahrenheit and Celsius temperatures are equal. What number is it?