Unit 7 Exponential Functions. Name: Period:

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1 Unit 7 Exponential Functions Name: Period: 1

2 AIM: YWBAT evaluate and graph exponential functions. Do Now: Your soccer team wants to practice a drill for a certain amount of time each day. Which plan will give your team more total practice time over 4 days? 8 days? Plan A: 5 minutes today and then 1 minute more each day than the previous day. Plan B: 1 minute today and then twice as much time each day as the previous day. Complete a chart and graph for each plan. Plan A: Day Practice Time Plan B: Day Practice Time Write each plan as a sequence where n represents the day and A(n) represents the practice time: Plan A:,,,,, Plan B:,,,,, Plan represents an Sequence, because the difference between each consecutive term in the sequence is. Write an explicit formula to represent the sequence. Plan represents a Sequence. In a geometric sequence the difference between the consecutive terms is a, r. The explicit formula would be A(n) = A(1)r (n 1) In this example: A(n) = A geometric sequence is an example of an function. An Exponential Function is an equation in which the variable is an. For example: y = a x where a is any real number. 2

3 Use your calculator to help complete the chart and sketch each of the following graphs: y = 2 x x y y = ( 1 2 )x y = -(2 x ) x y x y The graph of the function y = a x is when a > 1. The graph of the function y = a x is when 0 < a < 1. The graph of the function y = -a x is the of the graph of the function y = a x. The x-axis is an because the graph of the function will approach the x-axis, but will cross it. The value of an exponential function is zero. All exponential functions of the form y = a x will cross through the point. Why? The place where a graph crosses the y-axis is called the. 3

4 Sketch the graph: y = 4 x over the domain [-2, 2] Steps: 1. Enter the equation in the calculator 2. Access the table 3. Copy the values from the table for the given interval 4. Plot at least 5 of the points in the table 5. Connect with a SMOOTH curve Sketch the graph of y = 3(2) x. How does this function compare to y = 2 x? What is the equation of the asymptote? Where does the graph cross the y-axis? 4

5 Homework: Exponential Functions Sketch each of the following: y = ( 2 5 )x over the domain [-3, 3] Is this function increasing or decreasing? What are the coordinates of the y-intercept? Sketch each of the following: y = 1.25 x over the domain [0, 10] Is the function increasing or decreasing? What are the coordinates of the y-intercept? 5

y = 2(3 x ) over the domain of all real numbers. Is this function increasing or decreasing? What are the coordinates of the y-intercept? What is the equation of the asymptote?

6 y = 2(3 x ) over the domain of all real numbers. Is this function increasing or decreasing? What are the coordinates of the y-intercept? What is the equation of the asymptote? Identify each of the following sequences as ARITHMETIC, GEOMETRIC, or NEITHER. If the sequence is arithmetic state the common difference, d. If it is geometric stat the common ratio, r. 1) 2, 4, 6, 8, 5) 1, 1, 2, 3, 5, 2) 2, 4, 8, 16, 6) 4, -8, 16, -32, 3) 2, 4, 7, 11, 7) 1, 0.5, 0.25, 0.125, 4) 224, 112, 56, 28, 8) 1, 4, 9, 16, 6

7 Exponential Growth and Decay AIM: YWBAT use exponential functions to model problem solving situations of exponential growth and decay. Do Now: You would like to buy a new iphone and will need $100 to make the purchase. You ask your parents if they will give you the money you need a week from today. They say that they will not. So instead you ask them if you could have $1 today, $2 tomorrow, $4 the next day and so on (doubling every day) until a week from today. They agree to the second plan. Do you think your parents made a wise deal? Complete this sequence: 1, 2, 4,,,,. This is an example of a sequence. The common is. Complete the chart below: Day Amount 1 $1 2 $1(2) = $ The explicit formula is: A(n) = If you were to continue to collect money according to this plan, how much would you collect on the 30 th day? Convert each of the following to decimals: To change a percent to a decimal, divide by. Move the decimal point places. 1) 3% 2) 125% 3) 20% 4) 2.6% Convert each of the following to a percent: To change a decimal to a percent, multiply by. Move the decimal places. 5) ) 2.4 7) 0.6 8)

Complete each of the following statements: 1. A shirt is on sale for 15% off of the original price. The person buying the shirt pays % of the original price. 2. The tax rate is 8.75%.

8 Complete each of the following statements: 1. A shirt is on sale for 15% off of the original price. The person buying the shirt pays % of the original price. 2. The tax rate is 8.75%. You pay % of the price for the item including tax. 3. The population increases 4% per year. Next year, the population will be % of this year s population. 4. The value of a car decreases by 20% each year. Next year, the value of the car will be % of it s value this year. Interest Formula Alec puts $100 in a CD that pays 2% interest per year. How much interest does the account earn in one year? Simple Interest: Compound Interest: The compound interest formula can be modeled by an increasing exponential function and is an example of exponential growth: 8

If $2000 is invested in the account at that rate, find the amount of money in the account after 10 years.

9 There are other situations that can be modeled by decreasing exponential functions, this is known as. Practice: 1) A bank is advertising a rate of 5% interest compounded annually. If $2000 is invested in the account at that rate, find the amount of money in the account after 10 years. 2) The population of a town is decreasing at a rate of 2.5% per year. If the population in the year 2000 was 28,000, what will be the expected population in 2015 if this rate of decrease continues. Give your answer to the nearest thousand. 9

10 Homework: Growth and Decay Write a formula that can be used to model each situation and then use it to solve the problem. 1) In 2010, the population of a city was 25,000. The population increased by 20% in each of the next three years. If this rate of increase continues, what will be the population of the city in 2015? 2) Alberto invested $5000 at 6% interest compounded annually. What will be the value of Alberto s investment after 8 years? 3) Mrs. Rubac has a trust fund from which she withdraws 5% each year. If the fund has a value of $50,000 this year, what will be the value of the fund after 10 years? 4) Hailey has begun a fitness program. The first week she ran 1 mile every day. Each week she increases the amount that she runs each day by 20%. In week 10, how many miles does she run each day? (Round to the nearest mile.) 5) Alex received $75 for his birthday. In the first week after his birthday, he spent one-third of the money. In the second week, and each of the following weeks, he spent one-third of the money he had left. How much money will Alex have left after 5 weeks? 10

Exponential Growth and Decay

Exponential Growth and Decay Identifying Exponential Growth vs Decay A. Exponential Equation: f(x) = Ca x 1. C: COEFFICIENT 2. a: BASE 3. X: EXPONENT B. Exponential Growth 1. When the base is greater than