Combined: Linear and Exponential Modeling
Linear FunctionExponential Functionf(x) = mx + b????or f(x) = m(x – x1) + y1f(x) = a · bxb is the starting value, m is the rate or the slope.m is positive for growth, negative for decay.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. exponentialIn 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 + bf(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.Exercise 1: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 = __________________. Exercise 2:Decide whether the table represents a linear or exponential function. Circle either linear or exponential. a.x01234567y2581114172023Linear or exponential? b.x01234567y3612244896192384Linear or exponential? c.x01234567y1052.51.25.625.3125.15625.078125Linear or exponential? Exercise 3: Evan and Kayla were given the information shown below about the bacteria growing in a Petri dish in their biology class.Number of hours (x)345678Number of Bacteria, B(x)3504506007809001200Kayla wants to model this information with a linear function. Evan wants to use an exponential function. Which model is the better choice? Explain why you chose this model.Exercise 4:Job A (DOCTOR) has a starting salary of $20,000 with yearly increases of $1500. Job B (TEACHER) has a starting salary of $10,000 with the salary doubling each year. After how many years will Job B exceed Job A? If you were to work for 3 years, which job would you choose? ................
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