Unit 9 B Linear Modeling Copyright 2015 2011

Linear Functions A linear function has a constant rate of change and a straight-line graph. § The rate of change is equal to the slope of the graph. § The greater the rate of change, the steeper the slope. § Calculate the rate of change by finding the slope between any two points on the graph. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 9, Unit B, Slide 2

Finding the Slope of a Line To find the slope of a straight line, look at any two points and divide the change in the dependent variable by the change in the independent variable. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 9, Unit B, Slide 3

Example 1 You hike a 3 -mile trail, starting at an elevation of 8000 feet. Along the way, the trail gains elevation at a rate of 650 feet per mile. The elevation along the trail (in feet) can be viewed as a function of distance walked (in miles). What is the domain of the elevation function? From the given data, draw a graph of a linear function that gives your elevation as you hike along the trail. Does this model seem realistic? Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 9, Unit B, Slide 4

Example 2 A small store sells fresh pineapples. Based on data for pineapple prices between $2 and $5, the storeowners created a model in which a linear function is used to describe how the demand (number of pineapples sold per day) varies with the price. For example, the point ($2, 80 pineapples) means that, at a price of $2 per pineapple, 80 pineapples can be sold on an average day. What is the rate of change for this function? Discuss the validity of this model. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 9, Unit B, Slide 5

The Rate of Change Rule The rate of change rule allows us to calculate the change in the dependent variable from the change in the independent variable. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 9, Unit B, Slide 6

Example 3 Using the linear demand function in Figure 9. 12, predict the change in demand for pineapples if the price increases by $3. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 9, Unit B, Slide 7

Equations of Lines General Equation for a Linear Function dependent variable = initial value + (rate of change independent variable) Algebraic Equation of a Line In algebra, x is commonly used for the independent variable and y for the dependent variable. For a straight line, the slope is usually denoted by m and the initial value, or y-intercept, is denoted by b. With these symbols, the equation for a linear function becomes y = mx + b. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 9, Unit B, Slide 8

Slope and Intercept For example, the equation y = 4 x – 4 represents a straight line with a slope of 4 and a y-intercept of – 4. As shown to the right, the y-intercept is where the line crosses the y-axis. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 9, Unit B, Slide 9

Varying the Slope The figure to the right shows the effects of keeping the same y-intercept but changing the slope. A positive slope (m > 0) means the line rises to the right. A negative slope (m < 0) means the line falls to the right. A zero slope (m = 0) means a horizontal line. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 9, Unit B, Slide 10

Varying the Intercept The figure to the right shows the effects of changing the y-intercept for a set of lines that have the same slope. All the lines rise at the same rate, but cross the y-axis at different points. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 9, Unit B, Slide 11

Example 4: Rain Depth Equation Use the function shown to the right to write an equation that describes the rain depth at any time after the storm began. Use the equation to find the rain depth 4 hours after the storm began. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 9, Unit B, Slide 12

Example 5 Alcohol is metabolized by the body in such a way that the blood alcohol content decreases linearly. A study by the National Institute on Alcohol Abuse and Alcoholism showed that, for a group of fasting males who consumed four drinks rapidly, the blood alcohol content rose to a maximum of 0. 08 g/100 m. L about an hour after the drinks were consumed. Three hours later, the blood alcohol content had decreased to 0. 04 g/100 m. L. Find a linear model that describes the elimination of alcohol after the peak blood alcohol content is reached. According to the model, what is the blood alcohol content five hours after the peak is reached? Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 9, Unit B, Slide 13

Example 6 Write an equation for the linear demand function in figure 9. 12. Then determine the price that should result in a demand of 80 pineapples per day. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 9, Unit B, Slide 14

Creating a Linear Function from Two Data Points Step 1: Let x be the independent variable and y be the dependent variable. Find the change in each variable between the two given points, and use these changes to calculate the slope, or rate of change. Step 2: Substitute the slope, m, and the numerical values of x and y from either point into the equation y = mx + b and solve for the y-intercept, b. Step 3: Use the slope and the y-intercept to write the equation in the form y = mx + b. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 9, Unit B, Slide 15

Example 7 Until about 1850, humans used so little crude oil that we can call the amount zero—at least in comparison to the amount used since that time. By 1960, humans had used a total (cumulative) of 600 billion cubic meters of oil. Create a linear model that describes world oil use since 1850. Discuss the validity of the model. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 9, Unit B, Slide 16

Example (cont) We know that population has risen exponentially since the mid-19 th century (see Unit 8 C), so a better model for oil consumption would be exponential instead of linear. The blue curve in Figure 9. 16 shows an exponential fit to the two data points; we will discuss the creation of exponential models in the next unit. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 9, Unit B, Slide 17