Linear Geometry

A (3, -1) and B (9, -10) A (-4, 5) and B (1, -5) Linear coord geometry KUS objectives BAT use linear graphs in modelling; Starter: Find the equation of the line joining these points: give your answer in the form 𝑎𝑥+𝑏𝑦=𝑐 A (3, -1) and B (9, -10) 3𝑥+2𝑦=7 A (-4, 5) and B (1, -5) 4𝑥+2𝑦=−6 5𝑥−3𝑦=−14 A (-1, 3) and B (2, 8)

We often make assumptions in the model to simplify the situation. Notes 1 A mathematical model is an attempt to represent a real life situation using mathematical ideas. We often make assumptions in the model to simplify the situation. Speed is constant One variable varies in direct proportion to another There is no air resistance A mathematical model is often used to make predictions which can then be compared to what actually happens in real life You will meet mechanical and statistical models of real life later in the course

Write a direct proportion equation connecting the two variables. 𝐸 (𝑐𝑚) 𝑚 (𝑔𝑟𝑎𝑚𝑠) 100 200 300 400 500 5 10 15 20 WB25 Spring (400, 20) The graph shows the extension 𝐸 of a spring when different masses, 𝑚, are attached to the end. Write a direct proportion equation connecting the two variables. Interpret your result. (0, 0) The equation has the form 𝐸=𝑘𝑚 Gradient is 𝑘= 20−0 400−0 = 1 20 So 𝐸 = 1 20 𝑚 Interpretation: Gradient 𝑘 represents the increase in extension when the mass increases by 1 gram

𝑂𝑖𝑙 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 (𝑚𝑖𝑙𝑙𝑖𝑜𝑛 𝑡𝑜𝑛𝑛𝑒𝑠) 500 1000 1500 2500 2000 5 10 15 20 𝐶𝑂2 Billion tonnes 𝑂𝑖𝑙 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 (𝑚𝑖𝑙𝑙𝑖𝑜𝑛 𝑡𝑜𝑛𝑛𝑒𝑠) 500 1000 1500 2500 2000 5 10 15 20 WB26 Carbon Dioxide The scatter graph shows the Oil production 𝑃 and carbon dioxide emissions, 𝐶, for various years since 1960 with a best fit line. X X X X X X X X X X (1500, 10) X X X Formulate a linear model linking 𝑃 and 𝐶 Give the relationship in the form 𝐶=𝑎𝑃+𝑏 Interpret the result and comment on the validity of the model for small values of 𝑃 (0, 3) Gradient is 𝑘= 10−3 1500−0 = 7 1500 So 𝐶 = 7 1500 𝑃+3 Interpretation: Gradient 7 1500 represents the increase of 𝐶𝑂2 in billions of tonnes when the oil production increases by 1 million tonnes. The model is not reliable for small values of 𝑃

One thing to improve is – KUS objectives BAT use linear graphs in modelling self-assess One thing learned is – One thing to improve is –

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