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LINEAR AUTOMOBILE DEPRECIATION Section 5-5 Linear Depreciation • Most cars will not be worth their purchase price as they get older • They depreciate in value over time • Lose value • Some collectible cars increase in value over time • Appreciate Linear Depreciation • The simplest form of depreciation is straight line depreciation • This means that the car loses the same amount of value each year • If we can determine the equation of this model, we can find the value of the car at any time in its lifespan Linear Depreciation • You purchase a car for $27,000. After some research, you learn that the car loses all of its marketable value after 12 years (it depreciates to a value of zero dollars 12 years after the purchase date). • Find the equation for this model. Linear Depreciation • Finding the equation for a straight line depreciation model: 1) Find the intercepts (0, max car value) and (max life span, 0) Linear Depreciation • You purchase a car for $27,000. After some research, you learn that the car loses all of its marketable value after 12 years (it depreciates to a value of zero dollars 12 years after the purchase date). Max car value: $27,000 Max lifespan: 12 years Intercepts: (0, 27,000) (12, 0) Linear Depreciation • Finding the equation for a straight line depreciation model: 1) Find the intercepts (0, 27,000) and (12, 0) 2) Calculate the slope of the line = 2 − 1 2 − 1 Linear Depreciation • You purchase a car for $27,000. After some research, you learn that the car loses all of its marketable value after 12 years (it depreciates to a value of zero dollars 12 years after the purchase date). Intercepts: (0, 27,000) (12, 0) 2 − 1 27,000 − 0 = =- 2,250 = 2 − 1 0 − 12 Linear Depreciation • Finding the equation for a straight line depreciation model: 1) Find the intercepts (0, 27,000) and (12, 0) 2) Calculate the slope of the line = −2,250 3) Substitute slope and y-intercept into the equation: y = mx + b Linear Depreciation • You purchase a car for $27,000. After some research, you learn that the car loses all of its marketable value after 12 years (it depreciates to a value of zero dollars 12 years after the purchase date). Intercepts: (0, 27,000) (12, 0) = −2,250 Y –intercept = 27,000 Y = - 2,250x + 27,000 Linear Depreciation • You purchase a car for $27,000. After some research, you learn that the car loses all of its marketable value after 12 years (it depreciates to a value of zero dollars 12 years after the purchase date). Y = - 2,250x + 27,000 How much is the car worth after 7 years? Y = - 2,250(7) + 27,000 = $11,250 LINEAR AUTOMOBILE DEPRECIATION Section 5-5 Linear Depreciation • Yesterday we created equations that modeled a straight line depreciation value of a car. 1) Find the intercepts (0, max car value) and (max life span, 0) 2 − 1 = 2) Calculate the Slope 2 − 1 3) Plug them into the equation of a line in slope- intercept form Y = mx + b Linear Depreciation • Write the straight line depreciation equation for a car that was purchased for $22,000 and totally depreciates after 11 years. Linear Depreciation • Today, we are going to take this one step further and graph the equations. 1) Choose an appropriate window for the axis 2) Plot the intercepts 3) Graph the line between the intercepts Linear Depreciation • Draw the graph of the straight line depreciation model for a car that was purchased for $22,000 and totally depreciates after 11 years. Y = -2,000x + 22,000 $30,000 $25,000 $20,000 $15,000 $10,000 $5,000 1 2 3 4 5 6 7 8 9 10 11 12 Linear Depreciation • Suppose you buy a car at a price of $27,600. According to research, similar cars have depreciated to zero value after 12 years. a) Create a straight line depreciation model for this car. b) Graph the line to this model. c) How much will the car be worth after 66 months? Linear Depreciation • Suppose you buy a car at a price of $27,600. According to research, similar cars have depreciated to zero value after 12 years. a) Create a straight line depreciation model for this car. Intercepts:(0, 27,600) (12, 0) Slope: 27,600 − 0 = - 2,300 0 − 12 Y-Intercept: 27,600 Equation: Y = - 2,300x + 27,600 Linear Depreciation • Suppose you buy a car at a price of $27,600. According to research, similar cars have depreciated to zero value after 12 years. b) Graph the line to this model. $30,000 $25,000 $20,000 $15,000 $10,000 $5,000 1 2 3 4 5 6 7 8 9 10 11 12 Linear Depreciation • Suppose you buy a car at a price of $27,600. According to research, similar cars have depreciated to zero value after 12 years. c) How much will the car be worth after 66 months? 66 Months = 5.5 years Y = - 2,300x + 27,600 Y = - 2,300(5.5) + 27,600 Y = $14,950 LINEAR AUTOMOBILE DEPRECIATION Section 5-5 Linear Depreciation • A car is purchased for $85,000. After researching the make and model of the car, it is determined that the car depreciates in a straight line model after 25 years. Write the straight line depreciation model for this situation. Linear Depreciation • A car is purchased for $85,000. After researching the make and model of the car, it is determined that the car depreciates in a straight line model after 25 years. Graph the equation from part a. Linear Depreciation • A car is purchased for $85,000. After researching the make and model of the car, it is determined that the car depreciates in a straight line model after 25 years. What is the value of the car after 10 years? Linear Depreciation • A car is purchased for $85,000. After researching the make and model of the car, it is determined that the car depreciates in a straight line model after 25 years. How long will it take the car to lose 25% of its original value? Linear Depreciation • A car is originally worth $34,450. It takes 13 years for the car to totally depreciate. Write the straight line depreciation model for this situation. Y = - 2,650x + 34,450 How long will it take the car to lose half its value? 6.5 years How long will it take for the car to be worth $10,000? 9.2 years Linear Depreciation • Automobile Expense Function • There are many different expenses that contribute to the running and upkeep of a car • For the purposes of this class, we are to only look at automobile expenses as: • The down payment made to purchase the car • The monthly payments you make to the lending institution Linear Depreciation • You buy a new car for $33,600. The car depreciates to zero after 10 years. Create a straight line depreciation model for this situation. Linear Depreciation • You make a $4,000 down payment, and pay $560 a month for 5 years to pay off your loan. Create an expense function for this model. Linear Depreciation • Graph both of these equations on the same axis. $35,000 $30,000 $25,000 $20,000 $15,000 $10,000 $5,000 10 20 30 40 50 60 70 80 90 100 110 120 Linear Depreciation • Linear Depreciation • Linear Depreciation • Linear Depreciation •