Math 8C Unit 6 – Day 4 Standards: Model exponential functions using tables, graphs, and equations, and translate between the three. Given a context, determine the growth/decay factor and rates. Use both recursive and explicit equations to notate exponential functions given a context. Identify key features of an exponential function. Use function notation and explain the relationship between domain and range.

Warm Up Identify the initial point and rate of change of each exponential function. 𝑓 𝑥 = 1 2 4 𝑥 𝐴= 1 2 𝐵=4 x f(x) 6 1 18 2 54 3 162 4 487 𝐴=6 𝐵=3

Must Do (3/1): Determine the initial point and growth/decay factor of the exponential relationship Write a function for the relationship. x f(x) 768 1 192 2 48 3 12 4 𝐼𝑃=768 𝐷𝐹= 1 4 𝐷𝑒𝑐𝑎𝑦 𝑅𝑎𝑡𝑒=75%

Must Do (3/3): Simplify 5 – 5(5 - x) = -2(x - 5) 2. (Challenge): Factor 12 𝑥 3 +18 𝑥 5 +60 𝑥 4 3. Find the domain and range of the exponential function: Domain: −∞<𝑥<∞ or All Real Numbers Range: 0<𝑦<∞ or 𝑦>0

Explicit vs. Recursive Equations Find the next three terms in the sequence 3, 6, 12, 24, … Now write a function rule for the relation. 48, 96, 192 𝑓 𝑥 =3 2 𝑥

Writing Recursive Rules Can we define the relationship another way? 3, 6, 12, 24, … Each term is twice the previous term… So 6=2∙3, and 12=2∙6, and 24=2∙12, … This precise explanation of term to term action is called recursion – the process of repeating items in a self-similar way.

Recursive Equations Recursive equations use adjacent terms to determine the previous or next term… The relation 3, 6, 12, 24, … is broken down as 6=2∙3 12=2∙6 24=2∙12 Let’s start with the rule 𝑛𝑒𝑥𝑡=2∙𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

Recursive Equations To keep track of which term we’re on, we use subscripts. So 𝑛𝑒𝑥𝑡=2∙𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 becomes 𝑎 𝑛 =2∙ 𝑎 𝑛−1 , 𝑎 0 =3

Example Find the first three terms of the sequence defined by 𝑎 𝑛 =3∙ 𝑎 𝑛−1 , 𝑎 0 =3 𝑎 1 = 𝑎 2 = 𝑎 3 = 9 27 81

You Try Find the first five terms of the sequence defined by 𝑎 𝑛 = 1 2 ∙ 𝑎 𝑛−1 , 𝑎 0 =896 𝑎 1 = 𝑎 2 = 𝑎 3 = 𝑎 4 = 𝑎 5 = 448 224 112 56 28

Example Write a recursive formula for the sequence 1, 4, 16, 64, … 𝑎 𝑛 =4∙ 𝑎 𝑛−1 , 𝑎 0 =1

You Try! Write a recursive formula for the sequence −2, −10, −50, −250, … 𝑎 𝑛 =5∙ 𝑎 𝑛−1 , 𝑎 0 =−2

Recap Recursive equations use only the previous term to get the next term. Recursive equations show the growth/decay factor, and initial point. 𝑎 𝑛 =𝑟∙ 𝑎 𝑛−1 , 𝑎 0 =𝐴 𝑟=𝐺𝑟𝑜𝑤𝑡ℎ 𝑜𝑟 𝐷𝑒𝑐𝑎𝑦 𝐹𝑎𝑐𝑡𝑜𝑟, 𝐴=𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑃𝑜𝑖𝑛𝑡

Example A rainforest loses a third of its woodland every year. If it started out at 3,306,744 square miles, how many square miles would be left after 10 years? Write an explicit and a recursive equation to model the loss of rainforest, then answer the question. 𝑓 𝑥 =3306744 2 3 𝑥 𝑎 𝑛 = 2 3 ∙ 𝑎 𝑛−1 , 𝑎 0 =3306744 𝑓 10 =57,344 𝑎 10 =57,344 57,344 square miles of rainforest left after 10 years

Example 𝑎 𝑛 = 1 2 ∙ 𝑎 𝑛−1 , 𝑎 0 =500 𝑎 1 =250 𝑎 5 =15.625 𝑎 2 =125 A mine worker discovers an ore sample containing 500 mg of radioactive material.  It is discovered that the radioactive material has a half life of 1 day.  Find the amount of radioactive material in the sample at the beginning of the 7th day. Write a recursive and explicit equation and find the first seven terms. 𝑎 𝑛 = 1 2 ∙ 𝑎 𝑛−1 , 𝑎 0 =500 𝑎 1 =250 𝑎 5 =15.625 𝑎 2 =125 𝑎 6 =7.8125 𝑎 3 =62.5 𝑎 7 =3.90625 𝑎 4 =31.25

You Try! 𝑎 𝑛 =(1+.10)∙ 𝑎 𝑛−1 , 𝑎 0 =75 or 𝑎 𝑛 =(1.10)∙ 𝑎 𝑛−1 𝑎 1 =82.5 The hot tub in your hotel suite is not hot enough! It’s supposed to be hot, right?  The hotel tells you that they will increase the temperature by 10% each hour.  If the current temperature of the hot tub is 75º F, what will be the temperature of the hot tub each hour for the next 3 hours (to the nearest tenth of a degree)? Define a recursive and explicit equation and find the first three terms. 𝑎 𝑛 =(1+.10)∙ 𝑎 𝑛−1 , 𝑎 0 =75 or 𝑎 𝑛 =(1.10)∙ 𝑎 𝑛−1 𝑎 1 =82.5 𝑎 2 =90.8 𝑎 3 =99.9

In Class Practice Recursive Rules

Must Do (3/7): For each problem, write an explicit and a recursive equation: 3 , 3 2 , 3 4 , 3 8 …… 33, 198, 1188, 7128,…….. Challenge! 3. 13.7, 20.55, 30.825, 46.238……..

Olympic Times Task Overall feedback: Some arguments: Much more specific, better analysis Missing labels on axes or lines, key on graph Need support for your arguments; detailed paragraph If using sources, MLA work-cited bibliography Some arguments: Missing data led to inaccuracies Line of best fit should be curved Limit on human speed (100 m in 9.48 seconds) Weather, altitude, training, wealth, clothes/shoes, drug use can all affect outcomes

Exit Ticket Suppose you drop a tennis ball from a height of 15 feet. After the ball hits the floor, it loses 15% of its previous height. How high will the ball rebound after its third bounce? Round to the nearest tenth. Write an explicit and a recursive equation to model the loss height, then answer the question.

Must Do (3/8): Write an explicit equation and solve 1. A Cap-Ed credit union checking account earns 2.5% every month. A Wells Fargo savings account earns 0.5% every month. If you started each account with $10,000, how much more would you make at Cap-Ed after 15 years? 2. Smoke detectors use a low-activity radioactive isotope, americium-241. It decays at a rate of 3.5% per month. If you start with 10 grams of the isotope, how much is left after 6 months? Cap-Ed: $851,717.89 Wells Fargo: $24,540.94 Difference: $827,176.95 8.08g

1. A Cap-Ed credit union checking account earns 2. 5% every month 1. A Cap-Ed credit union checking account earns 2.5% every month. A Wells Fargo savings account earns 0.5% every month. If you started each account with $10,000, how much more would you make at Cap-Ed after 15 years? Cap-Ed: $851,717.89 Wells Fargo: $24,540.94 Difference: $827,176.95

2. Smoke detectors use a low-activity radioactive isotope, americium-241. It decays at a rate of 3.5% per month. If you start with 10 grams of the isotope, how much is left after 6 months? 8.08g

Must Do (3/9): Simplify: (−4 𝑥 −2 𝑦 3 ) −2 (−2 𝑥 8 𝑦 −6 ) 3 Find the domain and range

Must Do (3/10): Write a recursive and an explicit equation for each: -3, -5.7, -10.83, -20.577…...... 6, 4.5, 3, 1.5….... Write an explicit equation and solve: 3. A tool & die business purchased a piece of equipment of $250,000. The value of the equipment depreciates at a rate of 12% each year. What is the value after 5 years?

Make an equation from a graph Steps: 1. Make an input/output table of 2-4 points from least to greatest in x-value 2. Make a 3rd column to adjust for the shift in the graph (±C)......if necessary 3. Find the growth/decay factor (r) 4. Find the y-intercept, or A (y value when x=0)

Make an equation from a graph y=3(2^x)+3

Exit Ticket (3/10) Write an explicit equation and solve: This year, an estimated 4,324,000 people in this country are illiterate. With new incentives and funding, the country is hoping to cut that number by 21% every year. If this trend holds, how many people will be illiterate in the year 2035? (round to nearest one) y=4324000(0.79^x) 49,068 people in 2035