Page 1 of 13 Linear, Quadratic, and Exponential Models Attendance Problems 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 2000, there were 4600 people. 2. Write an exponential decay function to model this situation. 3. Predict the population in 2020. I can compare linear, quadratic, and exponential models. Given a set of data, I can decide which type of function models the data and write an equation to describe the function. Common Core CCSS.MATH.CONTENT.HSF.LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. CCSS.MATH.CONTENT.HSF.LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table). CCSS.MATH.CONTENT.HSA.CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Page 2 of 13 Look at the tables and graphs. The data show three ways you have learned that variable quantities can be related. The relationship shown is linear. Look at the tables and graphs below. The data show three ways you have learned that variable quantities can be related. The relationship shown is quadratic. Look at the tables and graphs. The data show three ways you have learned that variable quantities can be related. The relationship shown is exponential.
! Algebra Page 3 of 13 People often gather data and then must decide what kind of relationship (if any) they think best describes their data. Video Example 1. Graph each data set. Which kind of model best describes the data? A. B.
Algebra 1 Graphing Data to Choose a Model Graph each data set. Which kind of model best describes the data? A Time (h) 0 1 2 3 Bacteria 10 20 40 80 Page 4 of 13 Plot the data points and connect them. The data appear to be exponential. Graph each data set. Which kind of model best describes the data? B C 0 5 10 15 20 F 32 41 50 59 68 Plot the data points and connect them. The data appear to be linear.
Page 5 of 13 Example 1. Graph each data set. Which kind of model best describes the data? A. Time(h) Bacteria 0 24 1 96 2 384 3 1536 4 6144 B. x y 3 14 2 9 1 6 0 5 1 6 2 9 3 14
Page 6 of 13 4. Guided Practice. Graph the data set. Which kind of model best describes the data? x y 3 0.30 2 0.44 0 1 1 1.5 2 2.25 3 3.38 Another way to decide which kind of relationship (if any) best describes a data set is to use patterns. Video Example 2. Look for a pattern in each data set to determine which kind of model best describes the data. A. B.
Algebra Page 7 of 13 2 Using Patterns to Choose a Model Look for a pattern in each data set to determine which kind of model best describes the data. A + 100 + 100 + 100 Height of Bridge Suspension Cables Cable s Distance from Tower (ft) Cable s Height (ft) 0 400 100 256 200 144 300 64-144 - 112-80 + 32 + 32 For every constant change in distance of +100 feet, there is a constant second difference of +32. The data appear to be quadratic. B + 1 + 1 + 1 Value of a Car Car s Age (yr) Value ($) 0 20,000 1 17,000 2 14,450 3 12,282.50 0.85 0.85 0.85 For every constant change in age of +1 year, there is a constant ratio of 0.85. The data appear to be exponential.
Page 8 of 13 Example 2. Look for a pattern in each data set to determine which kind of model best describes the data. A. Height of golf ball Time (s) Height (ft) 0 4 1 68 2 100 3 100 4 68 B. Money in CD Time (yr) Amount ($) 0 1000.00 1 1169.86 2 1368.67 3 1601.04 6. Guided Practice. Look for a pattern in the data set {( 2, 10), ( 1, 1), (0, 2), (1, 1), (2, 10)} to determine which kind of model best describes the data. After deciding which model best fits the data, you can write a function. Recall the general forms of linear, quadratic, and exponential functions.
Page 9 of 13 Video Example 3. Use the data in the table to describe how the ladybug population is changing. Then write a function that models the data. Use your function to predict the ladybug population after two years.
Algebra Page 10 of 13 3 Problem-Solving Application Use the data in the table to describe how the ladybug population is changing. Then write a function that models the data. Use your function to predict the ladybug population after one year. Ladybug Population Time (mo) Ladybugs 0 10 1 30 2 90 3 270 1 Understand the Problem The answer will have three parts a description, a function, and a prediction. 2 Make a Plan Determine whether the data is linear, quadratic, or exponential. Use the general form to write a function. Then use the function to find the population after one year. 3 Solve Step 1 Describe the situation in words. Ladybug Population + 1 + 1 + 1 Time (mo) Ladybugs 0 10 1 30 2 90 3 270 3 3 3 Each month, the ladybug population is multiplied by 3. In other words, the population triples each month. Step 2 Write the function. There is a constant ratio of 3. The data appear to be exponential. y = a b x Write the general form of an exponential function. y = a (3) x 10 = a (3) 0 10 = a (1) 10 = a y = 10 (3) x Substitute the constant ratio, 3, for b. Choose an ordered pair from the table, such as (0, 10). Substitute for x and y. Simplify. 3 0 = 1 The value of a is 10. Substitute 10 for a in y = a (3) x. Step 3 Predict the ladybug population after one year. y = 10 (3) x Write the function. = 10 (3) 12 = 5,314,410 Substitute 12 for x (1 year = 12 mo). Use a calculator. There will be 5,314,410 ladybugs after one year. 4 Look Back You chose the ordered pair (0, 10) to wr ite the function. Check that ever y other ordered pair in the table satisfies your function. y = 10 (3) x 30 10 (3) 1 30 10 (3) 30 30 y = 10 (3) x 90 10 (3) 2 90 10 (9) 90 90 y = 10 (3) x 270 10 (3) 3 270 10 (27) 270 270
Page 11 of 13 Example 3. 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. Time (Days) E-mail forwarding Number of People Who Received the E-mail 0 8 1 56 2 392 3 2744 Remember! When the independent variable changes by a constant amount, linear functions have constant first differences. quadratic functions have constant second differences. exponential functions have a constant ratio. Teacher: Why are you so dressed up? Student: You said we d be modeling today.
Page 12 of 13 7. Guided Practice. 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. Summary. Which kind of model best describes each set of data? 8. 9.
Page 13 of 13 10. 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. 9-4 Assignment (p 653) 8-14, 23-26.