Lines - CHAPTER 2 Lines and Graphs Almost everything in...

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Unformatted text preview: CHAPTER 2 Lines and Graphs Almost everything in this chapter is revision from GCSE maths. It re- minds you how to draw graphs, and focuses in particular on straight line graphs and their gradients . We also look at graphs of quadratic functions, and use graphs to solve equations and inequalities. An im- portant economic application of straight line graphs is budget con- straints . — ./ — 1. The Gradient of a Line A is a point with co-ordinates (2 , 1); B has co-ordinates (6 , 4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x y 1 2 3 4 5 6-1 1 2 3 4 5-1 A B 4 3 When you move from A to B , the change in the x-coordinate is Δ x = 6- 2 = 4 and the change in the y-coordinate is Δ y = 4- 1 = 3 The gradient (or slope) of AB is Δ y divided by Δ x : Gradient = Δ y Δ x = 3 4 = 0 . 75 (The symbol Δ, pronounced “delta”, denotes “change in”.) It doesn’t matter which end of the line you start. If you move from B to A , the changes are negative, but the gradient is the same: Δ x = 2- 6 =- 4 and Δ y = 1- 4 =- 3, so the gradient is (- 3) / (- 4) = 0 . 75. There is a general formula: The gradient of the line joining ( x 1 ,y 1 ) and ( x 2 ,y 2 ) is: Δ y Δ x = y 2- y 1 x 2- x 1 23 24 2. LINES AND GRAPHS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x y 1 2 3 4 5 6-1 1 2 3 4 5-1 C D 2 4 Here the gradient is negative. When you move from C to D : Δ x = 4- 2 = 2 Δ y = 1- 5 =- 4 The gradient of CD is: Δ y Δ x =- 4 2 =- 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x y a b c d In this diagram the gradient of line a is positive, and the gradient of b is negative: as you move in the x-direction, a goes uphill, but b goes downhill. The gradient of c is zero. As you move along the line the change in the y-coordinate is zero: Δ y = 0 The gradient of d is infinite. As you move along the line the change in the x-coordinate is zero (so if you tried to calculate the gradient you would be dividing by zero). Exercises 2.1 : Gradients (1) (1) Plot the points A (1 , 2) ,B (7 , 10) ,C (- 4 , 14) ,D (9 , 2) and E (- 4 ,- 1) on a diagram. (2) Find the gradients of the lines AB,AC,CE,AD . 2. LINES AND GRAPHS 25 2. Drawing Graphs The equation y = 0 . 5 x +1 expresses a relationship between 2 quantities x and y (or a formula for y in terms of x ) that can be represented as a graph in x- y space. To draw the graph, calculate y for a range of values of x , then plot the points and join them with a curve or line....
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