2-4: Using Linear Models

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 12 t h 50 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? (0,12) (2,112) 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 ModelsDay 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, positivecorrelation correlation

2-4: Using Linear Equations • weak, negative strong, negative nocorrelation 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.

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. 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 originalpoints (unless they happento fall on the line) • Equation: y-intercept ≈ 4100 slope ≈ 1300 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) + 4100y = 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

2-4: Using Linear Models