Chapter 9: Modeling Our World Lecture notes Math 1030 Section BSection B.1: Linear FunctionsLinear model and linear functionIf the graph of a model is a straight line, the model is called linear and the function is called linear function.A linear function has a constant rate of change and a straight line graph. For all linear functions the rate ofchange is equal to the slope of the graph which is defined as the amount that the graph rises vertically fora given distance that it runs horizontally:rate of change = slope =change in dependent variablechange in independent variableThe greater the rate of change, the steeper the slope.1Chapter 9: Modeling Our World Lecture notes Math 1030 Section BEx.1You hike a 3-mile trail, starting at an elevation of 8000 feet. Along the way, the trail gains elevation ata rate of 650 feet per mile. What is the domain for the elevation function? From the given data, drawa graph of a linear function that could represent a model for your elevation as you hike along the trail.Does this model seem realistic?This model assumes that elevation increases at a constant rate along the entire 3-mile trail. While an eleva-tion change of 650 feet per mile seems reasonable as an average, the actual rate of change probably variesfrom point to point along the trail. Thus, the model’s predictions are likely to be reasonable estimates,rather than exact values, of your elevation at different points along the trail.2Chapter 9: Modeling Our World Lecture notes Math 1030 Section BEx.2A small stores sells fresh pineapples. Based on data for pineapple prices between $2 and $5, the storeown-ers created a model in which a linear function is used to describe how the demand (number of pineapplessold per day) varies with the price. The points ($2, 80 pineapples) and ($5, 50 pineapples) are in the graph.Draw the graph. What is the rate of change for this function? Discuss the validity of this model.This model seems reasonable within the domain for which the storeowners gathered data. Outside thisdomain, the model’s predictions probably are not true. For example, the model predicts that the storecould sell one pineapple per day at a price of $9.90, but could never sell a pineapple at a price of $10. Onthe other hand, the model predicts that the store could sell only 100 pineapples if they were free. This modelis useful only in a limited domain.3Chapter 9: Modeling Our World Lecture notes Math 1030 Section BThe Change in the Dependent VariableChange in dependent variableThe rate of change rule allows us to calculate the change in the dependent variable from the change in theindependent variable:change in dependent variable = rate of change × change in independent variableEx.3Using the linear demand in Example 2, predict the change for pineapples if the price increases by $3.4Chapter 9: Modeling Our World Lecture notes Math 1030 Section BSection B.2: General Equation for a Linear FunctionEquation for a linear functionThe general equation for a linear function ischange in dependent variable = initial value + rate of change × independent variableEx.4Suppose your job is to oversee an automated assembly line that manufactures computer chips. You arriveat work one day to find a stock of 25 chips that were produced during the night. If chips are producedat a constant rate of 4 chips per hour, how large is the stock of chips at any particular time during yourshift?5Chapter 9: Modeling Our World Lecture notes Math 1030 Section BThe Algebraic Equation of a LineAlgebraic equation of a lineIn algebra, x is used for the independent variable and y for the dependent variable. For a straight line, theslope is denoted by m and the initial value, or y-intercept, is denoted by b. The equation of a linear functionisy = mx + bThe equation of a general line isAx + By + C = 0If B = 0, then the line isx = −CAthat is, x = b (for example, x = 3).If B 6= 0, then the line can be written asy = −ABx −CBthat is, y = mx + b.Ex.5Draw the lines(1) x = 4(2) y = 2x + 1(3) y = 3x(4) y = −x − 1(5) y = 26Chapter 9: Modeling Our World Lecture notes Math 1030 Section BEx.6 Alcohol Methabolism.Alcohol is metabolized by the body in such a way that the blood alcohol content (BAC) decreases linearly.A study by the National Institute on Alcoholic Abuse and Alcoholism showed that, for a group of fastingmales who consumed four drink rapidly, the blood content rose to a maximum of0.8g100mLabout an hourafter the drinks were consumed. Three hours later, the blood alcohol content had decreases to0.04g100mL. Finda 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?7Chapter 9: Modeling Our World Lecture notes Math 1030 Section BEx.7Write an equation for the linear demand function in Example 2. Then determine the price that shouldresult in a demand of 80 pineapples per day.8Chapter 9: Modeling Our World Lecture notes Math 1030 Section BLinear Functions from Two Data PointsHow to find a linear function from two data pointsSuppose we have two data points and we want to find a linear function that fits them. We can find theequation for this linear function by using the two data points to determine the rate of change and the initialvalue for the function in 3 steps:Step 1: Let x be the independent variable and y be the dependent variable. Find the change in each variablebetween the two given points, and use these changes to calculate the slope (or rate of change):slope =change in ychange in xStep 2: Substitute this slope and the numerical values of x and y from either data point into the equationy = mx + b. You can then solve for the y-intercept b because it will be the only unknown in the equation.Step 3: Now use the slope and y-intercept to write the equation of the linear function in the form y = mx+b.Ex.8Until about 1850, humans used so little crude oil that we can call the amount zero, at least in comparisonto the amount used since that time. By 1960, humans had used a total of 600 billion cubic meters of oil.Create a linear model that describes world oil use since 1850. Discuss the validity of the