CHAPTER 2 : PROPERTIES OF STRAIGHT LINES. © John Wiley and Sons © John Wiley and Sons 2013 Essential Mathematics for Economics and Business, 4 th Edition

John Wiley and Sons 2013 Measure Slope and Intercept Figures 2.4. Different lines with (i) same slopes, (ii) same intercept.. How to draw a line, given slope and intercept. Worked Example 2.1 Figure 2.6 What is the equation of a line ? Write down the equation of a line, given its slope and intercept, Worked Example 2.2, Given the equation of a line, write down the slope and intercept Plot a line by joining the horizontal and vertical intercepts. Equations of horizontal and vertical lines, Figure 2.10

John Wiley and Sons 2013 Measuring Intercept The point at which a line crosses the vertical axis is referred as the ‘Intercept’ intercept = 2 intercept = 0 intercept =

John Wiley and Sons 2013 Measuring Slope Slope Line CD slope = Figure 2.4 D C A B Line AB slope = y increases by 0.5 when x increases by 1. y decreases by 1.2 when x increases by 1.

John Wiley and Sons 2013 Slope alone does not define a line Intercept alone does not define a line  Lines with same intercept but different slopes are different lines  Lines with same slope but different intercepts are different lines Intercept = 2 The slope of each line in this Figure is unity (1)

John Wiley and Sons A line is uniquely defined by both slope and intercept In mathematics, the slope of a line is referred to by the letter m In mathematics, the vertical intercept is referred to by the letter c Intercept, c = 2 slope, m = 1

John Wiley and Sons 2013 Draw the line, given slope, m =1: intercept, c = 2 Worked Example 2.1(b) 1. Plot a point at intercept = 2 2. From the intercept draw a line with slope = 1 by (a) moving horizontally forward by one unit and (b) vertically upwards by one unit 3. Extend this line indefinitely in either direction, as required The graph of the line whose intercept = 2, slope = 1 (0, 2) (1, 3) x Figure 2.6

John Wiley and Sons 2013 What does the equation of a line mean?  Consider the equation y = x  From the equation, calculate the values of y for x = 0, 1, 2, 3, 4, 5, 6.  The calculated points are given in the following table. x y  Plot the points as follows

John Wiley and Sons 2013 x y x y x y (0, 0) Plot the point x = 0, y = 0

John Wiley and Sons 2013 x y x y (0, 0) (1, 1) Plot the point x = 1, y = 1 y x

John Wiley and Sons 2013 x y x y (2, 2) (1, 1) (0, 0) Plot the point x = 2, y = 2 y x

John Wiley and Sons 2013 x y x y (2, 2) (1, 1) (0, 0) (3, 3) Plot the point x = 3, y = 3 y x

John Wiley and Sons 2013 x y x y y (2, 2) (1, 1) (0, 0) (3, 3) (4, 4) Plot the point x = 4, y = 4 x

John Wiley and Sons 2013 x y x y (2, 2) (1, 1) (0, 0) (3, 3) (4, 4) (5, 5) Plot the point x = 5, y = 5 y x

John Wiley and Sons 2013 x y x y (2, 2) (1, 1) (0, 0) (3, 3) (4, 4) (5, 5) (6, 6) Plot the point x = 6, y = 6 y x

John Wiley and Sons 2013 x y x y y (2, 2) (1, 1) (0, 0) (3, 3) (4, 4) (5, 5) (6, 6) Join the plotted points x

John Wiley and Sons 2013 x y x y (2, 2) (1, 1) (0, 0) (3, 3) (4, 4) (5, 5) (6, 6) The y co-ordinate = x co-ordinate, for every point on the line: Figure 2.9 The 45 o line, through the origin y x

John Wiley and Sons 2013 x y x y (2, 2) (1, 1) (0, 0) (3, 3) (4, 4) (5, 5) (6, 6) y = x is the equation of the line. Similar to Figure 2.9 y x

John Wiley and Sons 2013 Deduce the equation of the line, given slope, m = 1; intercept, c = 2 1. Start by plotting at least 2 points. Then observe the relationship between the x and y coordinate 2.Plot the intercept: x = 0, y = 2 (c =2) 3.Since slope = 1, move forward 1unit then up 1 unit. See Figure 2.6 This gives the point (x = 1, y = 3) 4.Map out further points, (2, 4) etc. in this way 5. Observe that value of the y co-ordinate is always “value of the x co-ordinate +2” Hence the equation of the line is y = x+ 2 6.That is, y = (1)x + (2), with m =1, c = 2 In general, y = mx + c is the equation of a line (0, 2) (1, 3) y (2, 4) Figure 2.6 x

John Wiley and Sons 2013 Deduce the equation of the line, given slope, m = 1; intercept, c = 2 Use Formula y = mx + c Since m = 1, c = 2 then y = mx + c y =(1)x + 2 y = x + 2 See Figure 2.6 (0, 2) (1, 3 ) y (2, 4) Figure 2.6 x

John Wiley and Sons 2013 The equation of a line Putting it another way: the equation of a line may be described as “ the formula that allows you to calculate the y co- ordinate when given the value of the x co-ordinate for any point on the line” Example 1. y = x “The y co-ordinate is equal to the x co-ordinate”. Slope, m = 1, intercept, c = 0. Example 2. y = x + 2 “The y co-ordinate is equal to the x co-ordinate plus 2”. Slope, m = 1, intercept, c = 2 The equation of a line may be written in terms of the two characteristics: m (slope) and c (intercept). y = mx + c

John Wiley and Sons 2013 Calculating the Horizontal Intercepts  Calculate the horizontal intercept for the line: y = mx + c  The horizontal intercept is the point where the line crosses the x - axis  Use the fact that the y co-ordinate is zero at every point on the x-axis.  Substitute y = 0 into the equation of the line  0 = mx + c  and solve for x:  0 = mx + c: therefore, x = -c/m  This is the formula for the horizontal intercept Line: y = mx + c (m > 0: c > 0) y = mx+ c Intercept =c Slope =m Horizontal intercept = - c/m 0, 0

John Wiley and Sons 2013 Determine the slope and intercepts for a line when the equation is given in the form: ax + by +d = 0  Rearrange the equation into the form y = mx + c.c.  Read off slope, m =  Read off intercept, c =  Horizontal intercept =  Example:4x + 2y - 8 = 0  Slope, m = -2: intercept, c = 4  Horizontal intercept =

John Wiley and Sons 2013 Plot the line 4x+2y - 8 = 0 by calculating the horizontal and vertical intercepts:  4x+2y - 8 = 0  Rearrange the equation into the form y = mx + c  y = -2x + 4  Vertical intercept at y = 4  Horizontal intercept at x = 2  Plot these points: see Figure 2.12  Draw the line thro’ the points Figure 2.12 y = -2x +4 y = 4 x = 2

John Wiley and Sons 2013 Equations of Horizontal and vertical lines:  The equation of a horizontal is given by the point of intersection with the y-axis  The equation of a vertical line is given by the point of intersection with the x -axis Figure 2.10