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1 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Chapter 2: The Straight Line and Applications u Measure Slope and Intercept Figures 2.4. Slide 2,4 u Different lines with (i) same slopes, (ii) same intercept. Slide 3. u How to draw a line, given slope and intercept. Worked Example 2.1 Figure 2.6, Slide 5 u What is the equation of a line ? Illustrated by Figure 2.9. Slides nos u Write down the equation of a line, given its slope and intercept, Worked Example 2.2, Figure 2.9 OR given the equation, write down the slope and intercept Slides no. 17, 18, 19 u Plot a line by joining the intercepts. Slide no.20, 21, 22 u Equations of horizontal and vertical lines, Figure 2.11: Slide no. 23

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2 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Measuring Slope and Intercept u The point at which a line crosses the vertical axis is referred as the Intercept u Slope = intercept = 2 intercept = 0 intercept = - 3 Line CD slope = Line AB slope = Figure 2.6

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3 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Slope alone or intercept alone does not define a line u Lines with same intercept but different slopes are different lines u Lines with same slope but different intercepts are different lines

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4 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd A line is uniquely defined by both slope and intercept u In mathematics, the slope of a line is referred to by the letter m u In mathematics, the vertical intercept is referred to by the letter c Intercept, c = 2 slope, m = 1

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5 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Draw the line, given slope =1: intercept = 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 which has intercept = 2, slope = 1 (0, 2) ( 1, 3) x Figure 2.6

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6 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd What does the equation of a line mean u Consider the equation y = x u From the equation, calculate the values of y for x = 0, 1, 2, 3, 4, 5, 6. u The points are given in the following table. x y u Plot the points as follows

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7 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd x y x y x y (0, 0) Plot the point x = 0, y = 0

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8 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd x y x y x y (0, 0) (1, 1) Plot the point x = 1, y = 1

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9 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd x y x y x y (2, 2) (1, 1) (0, 0) Plot the point x = 2, y = 2

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10 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd x y x y x y (2, 2) (1, 1) (0, 0) (3, 3) Plot the point x = 3, y = 3

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11 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd x y x y x y (2, 2) (1, 1) (0, 0) (3, 3) (4, 4) Plot the point x = 4, y = 4

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12 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd x y x y x y (2, 2) (1, 1) (0, 0) (3, 3) (4, 4) (5, 5) Plot the point x = 5, y = 5

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13 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd x y 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

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14 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd x y x y x y (2, 2) (1, 1) (0, 0) (3, 3) (4, 4) (5, 5) (6, 6) Join the plotted points

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15 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd x y 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

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16 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd x y 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

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17 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Deduce the equation of the line, given slope, m = 1; intercept, c = 2 1. Determine and plot at least 2 points: 2.Start at x = 0, y = 2 (intercept, c=2) 3.Since slope = 1, move forward 1unit then up 1 unit. See Figure 2.6 Hence the point (x = 1, y = 3) 4.Deduce further points in this way 5. Observe that value of the y co-ordinate is always (value of the x co-ordinate +2): Hence the equation y = x+ 2 6.That is, y = (1)x + 2 In general, y = mx + c is the equation of a line (0, 2) ( 1, 3) x y (2, 4) Figure 2.6

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18 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd 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 = 1x + 2 y = x + 2 See Figure 2.6 (0, 2) ( 1, 3) x y (2, 4) Figure 2.6

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19 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd The equation of a line u Putting it another way: u the equation of a line may be described as the formula that allows you to calculate the y co-ordinate for any point on the line, when given the value of the x co-ordinate. Example: u y = x is a line which has a slope = 1, intercept = 0 Example: u y = x + 2 is the line which has a slope = 1, intercept = 2 The equation of a line may be written in terms of the two characteristics, m (slope) and c (intercept). y = mx + c

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20 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Calculating the Horizontal Intercepts u Calculate the horizontal intercept for the line: y = mx + c u The horizontal intercept is the point where the line crosses the x -axis u Use the fact that the y co-ordinate is zero at every point on the x-axis. u Therefore, substitute y = 0 into the equation of the line 0 = mx + c u and solve for x: u 0 = mx + c: therefore, x = -c/m u This is the value of horizontal intercept Line: y = mx + c (m > 0: c > 0) y = mx+ c Intercept =c Slope =m Horizontal intercept = - c/m 0, 0

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21 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Determine the slope and intercepts for a line when the equation is given in the form: ax + by +d = 0 u Rearrange the equation into the form y = mx + c : u Slope = : intercept = u Horizontal intercept = Example:4x + 2y - 8 = 0 u Slope = -2: intercept = 4 u Horizontal intercept =

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22 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Plot the line 4x+2y - 8 = 0 by calculating the horizontal and vertical intercepts: u 4x+2y - 8 = 0 u Rearrange the equation into the form y = mx + c u y = -2x + 4 u Vertical intercept at y = 4 u Horizontal intercept at x = 2 (see previous slide) u Plot these points: see Figure 2.13 u Draw the line thro the points Figure 2.13

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23 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Equations of Horizontal and vertical lines: u The equation of a horizontal is given by the point of intersection with the y-axis u The equation of a vertical line is given by the point of intersection with the x -axis Figure 2.11

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