2. Write an exponential decay function to model this situation. Warm Up 1. Find the slope and y-intercept of the line that passes through (4, 20) and (20, 24). The population of a town is decreasing at a rate of 1.8% per year. In 1990, there were 4600 people. 2. Write an exponential decay function to model this situation. 3. Find the population in 2010.

Look at the tables and graphs below Look at the tables and graphs below. The data show two ways you have learned that variable quantities can be related. The relationship shown is linear.

Look at the tables and graphs below Look at the tables and graphs below. The data show two ways you have learned that variable quantities can be related. The relationship shown is quadratic.

Look at the tables and graphs below Look at the tables and graphs below. The data show two ways you have learned that variable quantities can be related. The relationship shown is exponential.

In the real world, people often gather data and then must decide what kind of relationship (if any) they think best describes their data.

Example 1A: Graphing Data to Choose a Model Which kind of model best describes the data? Time(h) Bacteria 24 1 96 2 384 3 1536 4 6144

Example 1B: Graphing Data to Choose a Model Which kind of model best describes the data? Boxes Reams of paper 1 10 5 50 20 200 500

x y –3 0.30 –2 0.44 1 1.5 2 2.25 3 3.38 Check It Out! Example 1a Which kind of model best describes the data? x y –3 0.30 –2 0.44 1 1.5 2 2.25 3 3.38

x y –3 –14 –2 –9 –1 –6 –5 1 2 3 Check It Out! Example 1b Which kind of model best describes the data? x y –3 –14 –2 –9 –1 –6 –5 1 2 3

We’re going to use our calculators to help us find models for the following data sets

We’re going to use our calculators to help us find models for the following data sets

After deciding which model best fits the data, you can write a function. Recall the general forms of linear, quadratic, and exponential functions.

Determine which model best fits the data and then find the equation for the model Height of golf ball Time (s) Height (ft) 4 1 68 2 100 3

Determine which model best fits the data and then find the equation for the model Money in CD Time (yr) Amount ($) 1000.00 1 1169.86 2 1368.67 3 1601.04

Determine which model best fits the data and then find the equation for the model Data (1) Data (2) –2 10 –1 1 2

Number of People Who Received the E-mail Example 3: Problem-Solving Application Use the data in the table to describe how the number of people changes. Then write a function that models the data. Use your function to predict the number of people who received the e-mail after one week. E-mail forwarding Time (Days) Number of People Who Received the E-mail 8 1 56 2 392 3 2744

Check It Out! Example 3 Use the data in the table to describe how the oven temperature is changing. Then write a function that models the data. Use your function to predict the temperature after 1 hour.

Lesson Quiz: Part I Which kind of model best describes each set of data? 1. 2.

Lesson Quiz: Part II 3. Use the data in the table to describe how the amount of water is changing. Then write a function that models the data. Use your function to predict the amount of water in the pool after 3 hours.

Comparing Functions

Example 2: Comparing Exponential Functions An investment analyst offers two different investment options for her customers. Compare the investments by finding and interpreting the average rates of change from year 0 to year 10.

Check It Out! Example 2 Compare the same investments’ average rates of change from year 10 to year 25.

Check It Out! Example 3 Students in an engineering class were given an assignment to design a parabola-shaped bridge. Suppose Rosetta uses y = –0.01x2 + 1.1x and Marco uses the plan below. Compare the two models over the interval [0, 20].

Example 4: Comparing Different Types of Functions A town has approximately 500 homes. The town council is considering plans for future development. Plan A calls for an increase of 50 homes per year. Plan B calls for a 5% increase each year. Compare the plans. Let x be the number of years. Let y be the number of homes. Write functions to model each plan

Check It Out! Example 4 Two neighboring schools use different models for anticipated growth in enrollment: School A has 850 students and predicts an increase of 100 students per year. School B also has 850 students, but predicts an increase of 8% per year. Compare the models. Let x be the number of students. Let y be the total enrollment. Write functions to model each school.

Lesson Quiz: Part III 3. A car manufacturer has 40 cars in stock. The manufacturer is considering two proposals. Proposal A recommends increasing the inventory by 12 cars per year. Proposal B recommends an 8% increase each year. Compare the proposals.