Modeling Our World 9A Discussion Paragraph 1 web 39. Daylight Hours 40. Variable Tables 1 world 41. Everyday Models 42. Functions and Variables Copyright © 2011 Pearson Education, Inc.

Unit 9B Linear Modeling Copyright © 2011 Pearson Education, Inc.

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. Walk through with your class several practical examples of the four types of slope (positive, negative, zero, undefined). Copyright © 2011 Pearson Education, Inc.

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 © 2011 Pearson Education, Inc.

Drawing a Linear Model CN (1a-b) 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. a. What is the domain for the elevation function? From the given data, draw a graph of linear function that could represent a model of your elevation as you hike along the trail. b. Does this model seem realistic? Copyright © 2011 Pearson Education, Inc.

A Price-Demand Function CN (2) A small store sells fresh pineapples. Based on data for pineapple prices between $2 and $7, 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. a. What is the rate of change for this function? b. Discuss the validity of this model. Copyright © 2011 Pearson Education, Inc.

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 © 2011 Pearson Education, Inc.

Change in Demand CN (3) 3. Using the linear demand functions in Figure 9.12 (demand, price of pineapples), predict the change in demand for pineapples if the price increases by $3. Copyright © 2011 Pearson Education, Inc.

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 © 2011 Pearson Education, Inc.

Slope and Intercept For example, the equation y = 4x – 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 © 2011 Pearson Education, Inc.

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 © 2011 Pearson Education, Inc.

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 © 2011 Pearson Education, Inc.

Rain Depth Equation CN (4) 4. 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 3 hours after the storm began. Since m = 0.5 and b = 0, the function is y = 0.5x. After 3 hours, the rain depth is (0.5)(3) = 1.5 inches. Copyright © 2011 Pearson Education, Inc.

Alcohol Metabolism CN (5a-b) Alcohol is metabolized by the body (through enzymes in the liver) 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 for fasting males who consumed four drinks rapidly, the blood alcohol content rose to a maximum of .08g/100mL about an hour after the drinks were consumed. Three hours later, the blood alcohol content had decreased to .04g/100mL. a. Find a linear model that describes the elimination of alcohol after the peak blood alcohol content is reached. b. According to the model, what is the blood alcohol content five hours after the peak is reached? Copyright © 2011 Pearson Education, Inc.

Price on Demand CN (6a-b) a. Write an equation for the linear demand function in Figure 9.12. b. Determine the price that should result in a demand of 8 pineapples per day. Copyright © 2011 Pearson Education, Inc.

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. 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 © 2011 Pearson Education, Inc.

Crude Oil Use Since 1850 CN (7a-b) 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. a. Create a linear model that describes world oil use since 1850. b. Discuss the validity of the model. Copyright © 2011 Pearson Education, Inc.

Quick Quiz CN (8) 8. Please answer the ten multiple choice questions from the Quick Quiz on p.531. Copyright © 2011 Pearson Education, Inc.

Homework 9B 9A Discussion Paragraph Class Notes 1-8 P. 532:1-10 1 web 50. Alcohol Metabolism 51. Property Depreciation 1 world 52. Linear Models 53. Nonlinear Models Copyright © 2011 Pearson Education, Inc.