Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 124.1 Linear Functions and Their PropertiesChapter 4 Section 1: Linear Functions and Their PropertiesIn this section, we will…Graph linear functionsIdentify the average rate of change of a linear functionDetermine whether a linear function is increasing, decreasing or constant over its domainSolve applications of linear functions4.1 Linear Functions and Their PropertiesThe average rate of change of a linear function is its slope.A linear function is a function of the form:It will have a slope of m and a y-intercept of (0, b)( )f x mx b= +The slope (rate of change) of a line gives the line’s steepness.1 21 2rise change in run change in y yymx x x-= = =-1 12 2for any two points ( , ) and ( , ) on the linex yx yWhy?4.1 Linear Functions and Their PropertiesA linear function is( )f x mx b= +…increasing over its domain if the slope is positive.…decreasing over its domain if the slope is negative.…constant over its domain if the slope is 0.What if the slope is undefned?Example: Indicate the slope and coordinate of the y-intercept of the given linear function, then graph the function.2( ) 43f x x=- +• slope:• y-intercept:• Determine the average rate of change of the function.•Determine whether the function is increasing, decreasing or constant.4.1 Linear Functions and Their PropertiesExample: Indicate the slope and coordinate of the y-intercept of the given linear function, then graph the function.( ) 2g x =-• slope:• y-intercept:• Determine the average rate of change of the function.•Determine whether the function is increasing, decreasing or constant.4.1 Linear Functions and Their PropertiesHow to Solve a Word Problem:Step 1: Read the problem until you understand it. • What are we asked to find? • What information is given?• What vocabulary is being used?Step 2: Assign a variable to represent what you are looking for. Express any remaining unknown quantities in terms of this variable.Step 3: Make a list of all known facts and form an equation or inequality to solve. It may help to make a labeled: diagram, table or chart, graphStep 4: SolveStep 5: State the solution in a complete sentence by mirroring the original question. Be sure to include units when necessary.Step 6: Check your result(s) in the words of the problem• Does your solution make sense?4.1 Linear Functions and Their PropertiesExample: The monthly cost C, in dollars, for international calls on Quahog Cellular Company is given by the function where x is the number of minutes used.• What is the cost if you talk on the phone for 50 minutes?• Suppose that your monthly bill is $29.32. How many minutes did you use?• Suppose that you budgeted $60 for the month for the phone. What is the maximum number of minutes you can talk.4.1 Linear Functions and Their Properties( ) 0.38 5,C x x= +The quantity supplied of a good or service is the amount of product a company is willing to supply at a given price.The quantity demanded of a good or service is the amount of product consumers are willing to purchase at a given price.4.1 Linear Functions and Their PropertiesEquilibrium occurs when supply = demandequilibrium priceequilibrium quantityExample: The quantity supplied S and the quantity demanded D of hot dogs at Quahog stadium are given by the following functions:Where p is the price of a hot dog.• Find the equilibrium price for hot dogs at the game. What is the equilibrium quantity. • Determine the prices for which quantity demanded is less than quantity supplied.• What will happen to the price of hot dogs if quantity demanded is greater than quantity supplied?4.1 Linear Functions and Their Properties( ) 2000 3000( ) 10,000 1000S x pD x p=- += -Depreciation is the reduction in value of an asset over its useful life.The simplest and most commonly used depreciation method, straight-line depreciation is calculated by taking the purchase or acquisition price of an asset subtracted by the salvage value divided by the total productive years the asset can be reasonably expected to benefit the company (called "useful life" in accounting jargon). We will assume there is no scrap-value.4.1 Linear Functions and Their PropertiesExample: Quahog Phone Company purchases a new company car for $28,000. If the company depreciates the vehicle using the straight-line method over 7 years, what will be the annual depreciation of the car?Example: Suppose that a company has just purchased a new computer system for $36,000. The company chooses to depreciate the computer using the straight-line method over three years.• Write the linear function that expresses the book-value V of the computer as a function of its age x.• Graph the linear function.• What is the book value of the computer after 2 years.• When will the computer system have a book value of $18,000.4.1 Linear Functions and Their PropertiesIndependent Practice You learn math by doing math. The best way to learn math is to practice, practice, practice. The assigned homework examples provide you with an opportunity to practice. Be sure to complete every assigned problem (or more if you need additional practice). Check your answers to the odd-numbered problems in the back of the text to see whether you have correctly solved each problem; rework all problems that are incorrect.Read pages 278-283Homework: pages 284-287 #15, 17, 19, 37, 39, 41, 474.1 Linear Functions and Their