Essential Question: How do you find the equation of a trend line? 2-4: Using Linear Models Essential Question: How do you find the equation of a trend line?

2-4: Using Linear Models You can write an equation to model real- world situations Example #1: Transportation Jacksonville, FL has an elevation of 12 ft above sea level. A hot-air balloon taking off from Jacksonville rises 50 ft/min. Write an equation to model the balloon’s elevation as a function of time. Graph the equation. Interpret the intercept at which the graph intersects the vertical axis.

2-4: Using Linear Models Jacksonville, FL has an elevation of 12 ft above sea level. A hot-air balloon taking off from Jacksonville rises 50 ft/min. Balloon’s elevation = rate • time + starting elevation Let h = the balloon’s current height Let t = time (in minutes) since the balloon lifted off = • + So an equation that models this data is: We’ll graph it on the next slide h 50 t 12 h = 50t + 12

2-4: Using Linear Models h = 50t + 12 Let’s choose two points If t = 0: h = 50(0) + 12 = 12 Use the point (0, 12) If t = 2 h = 50(2) + 12 = 112 Use the point (2, 112) What does the y-intercept (0, 12) represent? (2,112) (0,12) It represents the initial height of the balloon from sea level. It started 12 feet up to begin with.

2-4: Using Linear Models Your Turn: Suppose a balloon begins descending at a rate of 20 ft/min from an elevation of 1350 ft. Write an equation to model the balloon’s elevation as a function of time. What is true about the slope of the line? Graph the equation. Interpret the h-intercept.

2-4: Using Linear Models You can use two data points from a linear relationship (in point-slope form) to write a model. Example #2/3: Science A candle is 6 in tall after burning for 1h. After 3h, it is 5½ in tall. Write a linear equation to model the height y of the candle after burning x hours. In how many hours will the candle be 4 in tall?

2-4: Using Linear Models A candle is 6 in tall after burning for 1 h After 3 h, it is 5½ in tall. Write a linear equation to model the height y of the candle after burning x hours What are the two data points we have to use? What is the equation for point slope form? What do we have from that equation? What do we need? (1, 6) and (3, 5½) y – y1 = m(x – x1) y1 = 6 and x1 = 1

2-4: Using Linear Models Points: (1, 6) and (3, 5½) Find the slope: Find the equation in point-slope form y – y1 = m(x – x1) y – 6 = -¼(x – 1) y – 6 = -¼ x + ¼ y = -¼ x + 6 ¼

2-4: Using Linear Equations y = -¼ x + 6¼ In how many hours will the candle be 4 in tall? Recall back in our original problem, height is y Substitute 4 for y and solve for x 4 = -¼ x + 6¼ The candle will be 4 in tall after 9 hours -2¼ = -¼ x 9 = x

2-4: Using Linear Models Your Turn: y = -¼ x + 6¼ What does the slope -¼ represent? What does the y-intercept 6¼ represent? How tall will the candle be after burning for 11 hours? When will the candle burn out? The rate the candle burns down (¼ in per hour) The original height of the candle y = -¼(11) + 6¼ y = -2¾ + 6¼ = 3½ inches 0 = -¼ x + 6¼ -6¼ = -¼ x 25 = x About 25 hours for the candle to burn out

2-4: Using Linear Models Assignment Page 81 Problems 1 – 7 (all) Friday: Quiz Direct Variation (Last Thursday, Section 2-3) Absolute Value Functions/Graphs (Monday, Section 2-5) Families of Functions (Tuesday, Section 2-6) Using Linear Models (today)

2-4: Using Linear Models Day 2 Essential Question: How do you find the equation of a trend line?

2-4: Using Linear Equations Scatter Plot: a graph that relates two different sets of data by plotting the data as ordered pairs. You can use a scatter plot to determine a relationship between the data sets weak, positive strong, positive correlation correlation

2-4: Using Linear Equations weak, negative strong, negative no correlation correlation correlation Trend line: a line that approximates the relationship between the data sets of a scatter plot. You can use a trend line to make predictions. See the middle graph above for an example.

2-4: Using Linear Equations Example: Automobiles A woman is considering buying a 1999 used car for $4200. She researches prices for various years on the same model and records the data in the table below. Part A: Let x represent the model year (Use 1 for 2000, 2 for 2001 and so forth.) Let y be the price of the car. Draw a scatter plot. Decide whether a linear model is reasonable. Model Year 2000 2001 2002 2003 2004 Prices $5784 $6810 $8237 $9660 $10,948 $5435 $6207 $7751 $9127 $10,455

2-4: Using Linear Equations Example: Automobiles Is a linear model reasonable? Draw a trend line. Write the equation of the line and decide whether the asking price is reasonable. Model Year 2000 2001 2002 2003 2004 Prices $5784 $6810 $8237 $9660 $10,948 $5435 $6207 $7751 $9127 $10,455 Yes, see graph on left

2-4: Using Linear Equations After you’ve drawn your trend line, you can use the slope and y-intercept to determine an equation. You use the trend line, NOT any of the original points (unless they happen to fall on the line) Equation: slope ≈ 1300 y-intercept ≈ 4100 y = 1300x + 4100

2-4: Using Linear Equations Is a 1999 car for $4200 a reasonable price? The equation is y = 1300x + 4100 A 1999 car would represent the year x = 0 Remember, we started by using x = 1 for a 2000 car So a 1999 car would be fairly priced at: y = 1300(0) + 4100 y = 4100 A price of $4200 is a reasonable price

2-4: Using Linear Equations Your Turn: Graph each set of data. Decide whether a linear model is reasonable. If so, draw a trend line and write its equation. {(-7.5, 19.75), (-2, 9), (0, 6.5), (1.5, 3), (4, -1.5)} Done on the whiteboard

2-4: Using Linear Equations Assignment Page 81 Problems 8 – 11 Show your graphs, trend lines, and equations Tomorrow: Quiz Direct Variation (Last Thursday, Section 2-3) Absolute Value Functions/Graphs (Monday, Section 2-5) Families of Functions (Tuesday, Section 2-6) Using Linear Models (Wednesday, 1st part of 2-4) Today’s material will NOT be on the quiz Next Week Monday: Chapter 2 Preview Tuesday: Chapter 2 Review Wednesday: Chapter 2 Test