Math 1 Exponential Functions Unit 2018

Transcription

1 1 Math 1 Exponential Functions Unit 2018 Points: /10 Name: Graphing Exponential Functions/Domain and Range Exponential Functions (Growth and Decay) Tables/Word Problems Linear vs Exponential Functions Compound Interest Sequences Sequences Word Problems

2 2 Warm-up 1. 2.

3 3 DOMAIN AND RANGE REVIEW State the Domain and Range. Is it a function? Which graph would have a domain and range of all real numbers. Domain: Range: Function? Domain: Range: Function?

4 4 Characteristics of Exponential Functions Graphing Exponential Equations 1. f(x) = 1 3 x x f(x) Domain: Range:

5 5 Graphing Exponential Equations Graph the following functions. Use the table to identify points. 1. f(x) = 1 2 x 2. f(x) = 4 2 x x f(x) x f(x) Domain: Domain: Range: Range: Sketch the following exponential functions. Identify the y-intercept, domain, and range. Domain: Range: Domain: Range:

6 Sketch the following graphs using the Desmos Exponential Functions website. Be sure to identify the y-intercept of each graph. Try to make connections between the equations and the types of graphs you see. 1: f(x) = 2(3) x + 0 2: f(x) = 2(0.7) x Changes? 3: f(x) = 2(3) x + 2 4: f(x) = 2(3) x 3 Changes? Changes?

7 7 5: f(x) = 2(0.4) x + 3 6: f(x) = 2(0.4) x 3 Changes: Changes? So what have your figured out???

8 8 Systems of Linear and Exponential Functions. Remember the answer to a system of equations is the (s). How many solutions could a system that has both a linear and an exponential function? Keystrokes for Finding Intersection Points On A Calculator. Type equations into Y= Hit Graph Adjust Window (as needed) 2 nd, Trace 5 Intersect First curve? ENTER Second curve? ENTER Guess? (get close to intersection point) ENTER Find the solution to the given system. Round to the nearest tenths place. 1: y=5x+3 y=2 x 2: y= 1 2 x 2 y=2(0.8) x 3: y= 6x 2 y=3(5) x

9 9 WARM UP Solve the system. 1. y = 2(0.5) x and y = 1 x y = 3(2) x and y = 2x What is the solution to the equation y = 120? 5.

10 10 Exponential Growth Function f(x) = a(b) x + c Exponential Decay Functions f(x) = a(b) x + c a: a: If b > 1 If 0 < b < 1 Then b is called Then b = 1 + r and r is The graph is Then b is called Then b = 1 r and r is The graph is

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12 12 More Examples: 1. f(x) = 3(1.15) x Growth or Decay: Initial Value: Growth/Decay Factor: Growth/Decay Rate: 2. f(x) = (. 125) x Growth or Decay: Initial Value: Growth/Decay Factor: Growth/Decay Rate: For each function fill in the table: Growth or Decay: Growth or Decay: Growth or Decay: Growth or Decay: Initial Value: Initial Value: Initial Value: Initial Value: Growth/Decay Factor: Growth/Decay Factor: Growth/Decay Factor: Growth/Decay Factor: Growth/Decay Rate: Growth/Decay Rate: Growth/Decay Rate: Growth/Decay Rate: Writing Exponential Functions- Write an equation for the exponential function with the given y- intercept and rate. 1: (0, 4), 5% increase 2. y-intercept 6, 80% decrease 3: (0, 5), 1 (decrease) 4: initial value is 5, doubles 2

13 13 Warm up 1: 2: What is the rate of the function f(x) = 2(0.7) x 3: Given the function f(x) = 4 x Linear or Exponential (and how do you know)? y-intercept? What is the rate of change from x = -1 to x = 1?

14 14 Exponential Functions from Tables Read the following example. Determine the rate of change for the problems If all the in a table of values have a constant difference and all the have a constant ratio, then the table represents an EXPONENTIAL FUNCTION.

15 15 b. c. 2. Decide whether the table represents a linear or exponential function. Circle either linear or exponential. Then, write the function formula. a. x y Linear or exponential? y =. b. x y Linear or exponential? y =. c. x y Linear or exponential? y =.

16 16 d. x y Linear or exponential? y = e. x y Linear or exponential? y = f. x y Linear or exponential? y = g. x y Linear or exponential? y = Exponential Functions WORD PROBLEMS Example: There are about 250 mice in your attic. After hiring an exterminator, the population begins to decrease at a rate of 70% per week. How many mice will there be in 8 weeks (create a function to model this first)? 1. A zombie infection in Duluth High School grows by 15% per hour. The initial group of zombies was a group of 4 freshmen. How many zombies are there after 6 hours?

17 17 2. Ryan is saving for his college tuition. He has $2,550 in a savings account that pays 6.25% annual interest. How much will he have in his account after 5 and ½ years. 3. Bacteria in a dirty glass triple every hour. If there are 25 bacteria to start, how many in the glass after 1 day? 4. Phil purchases a used truck for $11,500. The value of the truck is expected to decrease by 20% each year. When will the truck first be worth less than $1,000?

18 18 Warm up 1. Linear or Exponential? Write the equation to model each table Solve the following in TWO ways: 2x (x + 6) = 3x

19 19 Linear vs. Exponential Word Problems Linear Function Exponential Function f(x) = mx + b or f(x) = m(x x 1) + y 1 b is the starting value, m is the rate or the slope. m is positive for growth, negative for decay. f(x) = a b x 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. Choosing linear vs. exponential In growth and decay problems (that is, problems involving a quantity increasing or decreasing), here s how to decide whether to choose a linear function or an exponential function. If the growth or decay involves increasing or decreasing by a fixed number, use a linear function. The equation will look like: y = mx + b f(x) = (rate) x + (starting amount). If the growth or decay is expressed using multiplication (including words like doubling or halving ) use an exponential function. The equation will look like: f(x) = (starting amount) (base) x. Decide whether the word problem represents a linear or exponential function. Circle either linear or exponential. Then, write the function formula. a) A library has 8000 books, and is adding 500 more books each year. Linear or exponential? y =. b) A gym s customers must pay $50 for a membership, plus $3 for each time they use the gym. Linear or exponential? y =. c) A bank account starts with $10. Every month, the amount of money in the account is tripled. Linear or exponential? y =. d) At the start of a carnival, you have 50 ride tickets. Each time you ride the roller coaster, you have to pay 6 tickets. Linear or exponential? y =. e) There are 20,000 owls in the wild. Every decade, the number of owls is halved. Linear or exponential? y =.

20 1. A science experiment involves periodically measuring the number of mold cells present on a piece of bread. At the start of the experiment, there are 50 mold cells. Each time a periodic observation is made, the number of mold cells triples. For example, at observation #1, there are 150 mold cells. a. Write a function formula equation (y = ) for the number of mold cells present, where x stands for the observation number. 20 b. Fill in the missing outputs of this table. x = observation number y = mold cell count c. Suppose that the mold begins to be visible as green coloration when the mold cell count exceeds 100,000. On which observation will this happen? 2. Julie gets a pre-paid cell phone. Initially she has a $40.00 balance on the phone. Each minute of talking costs $0.15. Let x stand for the amount of time in minutes that Julie has talked on the phone, and let f(x) stand for the remaining dollar value of the phone. a. Is f(x) a linear function or an exponential function? Explain how you know. b. Find a function formula equation f(x) = c. Find the value of f (0) and explain its meaning in terms of the cell phone. d. Find the value of f (100) and explain its meaning in terms of the cell phone.

21 21 Warm Up 1: You buy a used car for $3,000. The value of the car depreciates at a yearly rate of 8%. Write the equation to model the situation. Find the value of the car after 6 months. 2: A local business had a profit of $38,000 in 2001 that increases by 5.5% per year. Write the equation to model the situation. Find the profit of the business in Write a brief scenario that would model each equation. Calculating Compound Interest Compound Interest is calculated by the formula

22 22 Compound Interest Formula: 1. A sum of $1000 is invested at an interest rate of 12% per year. Find the amounts in the account after 3 years if interest is compounded annually, semiannually, quarterly, monthly, and daily. P = r = t = Compounded n Compute Amount after 3 yrs Annual Semiannual Quarterly Monthly Daily Try It. 1: 2:

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30 30 Sequences Word Problems 4. A geometric sequence can be represented by the exponential function f(x) = 400(12) x. In terms of the geometric sequence, explain what f(3) = 50 represents.

31 31 Determine if the following situations describe an arithmetic or geometric sequence and if they require a linear or exponential growth model? Write an explicit formula for the sequence that models the growth for each case. a. A savings account that starts with $5000 and receives a deposit of $825 per month. b. The value of a house that starts at $150,000 and increases by 1.5% per year. c. An alligator population starts with 200 alligators and every year, the alligator population is 7 9 of the previous year s population. d. The temperature increases by 2 every 30 minutes from 8:00 a.m. to 3:30 p.m. for a July day that has a temperature of 66 at 8:00 a.m.