Quadratic Function Model

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Presentation transcript:

Quadratic Function Model Fuel Efficiency Context

Quadratic Functions In times of increasing gas prices, many drivers become concerned about the fuel efficiency of their vehicles. The United States Department of Energy reports that although each vehicle reaches its optimal fuel economy at a different speed, gas mileage usually increases up to speeds near 45 mph and then decreases rapidly at speeds above 60 mph (Source: www.fueleconomy.gov).

Quadratic Functions Speed (MPH) 10 20 30 40 50 60 70 80 Fuel Economy (MPG) 17.6 24.1 28.6 31.1 31.7 30.2 26.7 21.3 Imagine a continuous function that models these data. With your right index finger, track the increasing speed in MPH. With your left index finger, track the behavior of the fuel economy in MPG. Then track them both simultaneously. Explain the behavior of the function in the context of the situation. 3

Quadratic Functions

Fuel Economy vs Speed 6.5 4.5 -2.0 2.5 -2.0 0.6 -1.9 -1.5 -2.1 -3.5 Speed (S ) (MPH) Fuel Economy Model (F ) (MPG) F  (F) 10 17.6 20 24.1 30 28.6 40 31.1 50 31.7 60 30.2 70 26.7 80 21.3 6.5 4.5 -2.0 2.5 -2.0 0.6 -1.9 -1.5 -2.1 -3.5 -2.0 -5.4 -1.9 5

Fuel Economy vs Speed 9.1 10.5 -2.0 8.5 -2.0 6.5 -2.0 4.5 -2.0 2.5 Speed (S ) (MPH) Fuel Economy Model (F ) (MPG) F  (F) 10 17.6 20 24.1 30 28.6 40 31.1 50 31.7 60 30.2 70 26.7 80 21.3 9.1 10.5 -2.0 8.5 -2.0 6.5 -2.0 4.5 -2.0 2.5 -2.0 0.6 -1.9 -1.5 -2.1 -3.5 -2.0 -5.4 -1.9 6

Fuel Economy vs Speed “Starting” at 9.1 MPG when speed is (hypothetically) 0 MPH Initial rate of change is somewhere between 10.5 and 8.5 MPG for an increase of 10 MPH. Let’s say 9.5 MPG per 10 MPH or 0.95 MPG/MPH This initial rate of change is changing (decreasing) by half of 2 MPG per 10 MPH for each increase of 10 MPH in speed. Let’s say -0.01 MPG/MPH for each increase of 1 MPH

Fuel Economy vs Speed Mike’s Sentence The hypothetical initial fuel economy is 9.1 MPG for a car that travels at 0 MPH. This value initially changes at a rate of 0.95 MPG per 1 MPH increase in speed. This rate of change decreases by a constant 0.02 MPG per MPH for each increase in 1 MPH. 8

Thinking about Mike’s Sentence Covariationally 2.5 4.5 6.5 8.5