Chapter 2: The Straight Line and Applications Measure Slope and Intercept Figures 2.4. Slide 2,4 Different lines with (i) same slopes, (ii) same intercept. Slide 3. How to draw a line, given slope and intercept. Worked Example 2.1 Figure 2.6, Slide 5 What is the equation of a line ? Illustrated by Figure 2.9. Slides nos. 6 - 16 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 Plot a line by joining the intercepts. Slide no.20, 21, 22 Equations of horizontal and vertical lines, Figure 2.11: Slide no. 23

Measuring Slope and Intercept The point at which a line crosses the vertical axis is referred as the ‘Intercept’ Slope = intercept = 2 4 3 2 intercept = 0 1 -4 -3 -2 -1 1 2 3 4 -1 -2 -3 intercept = - 3 Line AB slope = Line CD slope = -4 Figure 2.6 -5

Slope alone or 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

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

Draw the line, given slope =1: intercept = 2 Worked Example 2.1(b) The graph of the line which has intercept = 2, slope = 1 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 Figure 2.6 ( 1, 3) (0, 2) x

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 points are given in the following table. x 0 1 2 3 4 5 6 y 0 1 2 3 4 5 6 Plot the points as follows

x 0 1 2 3 4 5 6 y 0 1 2 3 4 5 6 1 2 3 4 5 6 x y (0, 0) Plot the point x = 0, y = 0

x 0 1 2 3 4 5 6 y 0 1 2 3 4 5 6 Plot the point x = 1, y = 1 6 5 4 3 y 2 (1, 1) 1 (0, 0) x 1 2 3 4 5 6

x 0 1 2 3 4 5 6 y 0 1 2 3 4 5 6 Plot the point x = 2, y = 2 6 5 4 3 y (2, 2) 2 (1, 1) 1 (0, 0) x 1 2 3 4 5 6

x 0 1 2 3 4 5 6 y 0 1 2 3 4 5 6 Plot the point x = 3, y = 3 6 5 4 (3, 3) 3 y (2, 2) 2 (1, 1) 1 (0, 0) x 1 2 3 4 5 6

x 0 1 2 3 4 5 6 y 0 1 2 3 4 5 6 Plot the point x = 4, y = 4 6 5 4 (4, 4) (3, 3) 3 y (2, 2) 2 (1, 1) 1 (0, 0) x 1 2 3 4 5 6

x 0 1 2 3 4 5 6 y 0 1 2 3 4 5 6 Plot the point x = 5, y = 5 6 (5, 5) 5 (4, 4) (3, 3) 3 y (2, 2) 2 (1, 1) 1 (0, 0) x 1 2 3 4 5 6

x 0 1 2 3 4 5 6 y 0 1 2 3 4 5 6 Plot the point x = 6, y = 6 6 (6, 6) (5, 5) 5 4 (4, 4) (3, 3) 3 y (2, 2) 2 (1, 1) 1 (0, 0) x 1 2 3 4 5 6

Join the plotted points x 0 1 2 3 4 5 6 y 0 1 2 3 4 5 6 Join the plotted points 6 (6, 6) (5, 5) 5 4 (4, 4) (3, 3) 3 y (2, 2) 2 (1, 1) 1 (0, 0) x 1 2 3 4 5 6

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

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

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

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

The equation of a line The equation of a line may be written in terms of the two characteristics, m (slope) and c (intercept) . y = mx + c Example: y = x is a line which has a slope = 1, intercept = 0 y = x + 2 is the line which has a slope = 1 , intercept = 2 Putting it another way: 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.

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. Therefore, 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 value of horizontal intercept Line: y = mx + c (m > 0: c > 0) y = mx + c Intercept = c Slope = m 0, 0 Horizontal intercept = - c/m

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: Slope = : intercept = Horizontal intercept = Example:4x + 2y - 8 = 0 Slope = -2: intercept = 4 Horizontal intercept =

Plot the line 4x+2y - 8 = 0 by calculating the horizontal and vertical intercepts: Rearrange the equation into the form y = mx + c y = -2x + 4 Vertical intercept at y = 4 Horizontal intercept at x = 2 (see previous slide) Plot these points: see Figure 2.13 Draw the line thro’ the points Figure 2.13

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 2.11