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

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

Measuring Slope and Intercept

The point at which a line crosses the vertical axis is referred as the Intercept
4 3
-4

Slope change in distance =

4 3 2 1

change in height

y x

intercept = 2 intercept = 0
-4 -3 -2 -1

0 -3 -2 -1 -1 -2 -3 -4 0 1 2 3 4

2
1

0
-1 -2

4
-5 -6

-3

intercept = - 3

-4 -5

Line AB slope =
2 4

Line CD slope =
6 5

Figure 2.6

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

Slope alone or intercept alone does not define a line

Lines with same intercept but different slopes are different lines
6 5 4

Lines with same slope but different intercepts are different lines
4 3 2 1

3
-4 -3 -2 -1

0 0 -1 -2 1 2 3 4

2 1

-3

0 -1 -1 -2 0 1 2 3 4 5 6

-4 -5

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

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
2 1

0 -1 0 1 2 3 4

Copyright2001 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 The graph of the line which has intercept = 2, slope = 1
Figure 2.6

(b) vertically upwards by one unit 3. Extend this line indefinitely in either direction, as required
-1

( 1, 3)
2

(0, 2)
1

0 0 1 2 3 4

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

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 y 0 1 2 3 4 Plot the points as follows

5 5

6 6

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

x y

0 0

1 1

2 2

3 3

4 4

5 5

6 6

Plot the point x = 0, y = 0

6

1 (0, 0) 0 1 2 3 4 5 6 x

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

x y

0 0

1 1

2 2

3 3

4 4

5 5

6 6

Plot the point x = 1, y = 1

6

3 y 2 (1, 1) 1 (0, 0) 0 1 2 3 4 5 6 x 0

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

x y

0 0

1 1

2 2

3 3

4 4

5 5

6 6

Plot the point x = 2, y = 2

6

3 y (2, 2)

2 (1, 1)

1 (0, 0) 0 1

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

x y

0 0

1 1

2 2

3 3

4 4

5 5

6 6

Plot the point x = 3, y = 3

6

4 (3, 3)

3 y (2, 2)

2 (1, 1)

1 (0, 0) 0 1

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

10

x y

0 0

1 1

2 2

3 3

4 4

5 5

6 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) 0 1

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

11

x y

0 0

1 1

2 2

3 3

4 4

5 5

6 6

Plot the point x = 5, y = 5

6 (5, 5)

5 (4, 4)

4 (3, 3)

3 y (2, 2)

2 (1, 1)

1 (0, 0) 0 1

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

12

x y

0 0

1 1

2 2

3 3

4 4

5 5

6 6

Plot the point x = 6, y = 6

6 (5, 5) (6, 6)

5 (4, 4)

4 (3, 3)

3 y (2, 2)

2 (1, 1)

1 (0, 0) 0 1

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

13

x y

0 0

1 1

2 2

3 3

4 4

5 5

6 6

Join the plotted points

6 (5, 5) (6, 6) 5 (4, 4)

4 (3, 3)

3 y (2, 2)

2 (1, 1)

1 (0, 0) 0 1

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

14

x y

0 0

1 1

2 2

3 3

4 4

5 5

6 6

The y co-ordinate = x co-ordinate, for every point on the line:

6 (5, 5) (6, 6)

5 (4, 4)

4 (3, 3)

3 y (2, 2)

2 (1, 1)

1 (0, 0) 0 1

Figure 2.9 The 45o line, through the origin

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

15

x y

0 0

1 1

2 2

3 3

4 4

5 5

6 6

y = x is the equation of the line.

6 (5, 5) (6, 6)

5 (4, 4)

4 (3, 3)

3 y (2, 2)

2 (1, 1)

Similar to Figure 2.9

0 0

(0, 0) 1 2 3 4 5 6 x

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

16

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
y
5

(2, 4)

( 1, 3)
2

(0, 2)
1

0 -1 0 1 2 3 4

Figure 2.6

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

17

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
-1

y
5

(2, 4)

( 1, 3)
2

(0, 2)
1

0 0 1 2 3 4

Figure 2.6

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

18

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 Example: y = x is a line which has a slope = 1, intercept = 0 Example: y = x + 2 is the line which has a slope = 1 , intercept = 2

= mx + c

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. 19

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

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

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

20

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:

ax by d 0 by d ax y d a x b b
a b

Example:4x + 2y - 8 = 0
4x 2 y 8 0 2 y 8 4x y 4 2x

Slope =

d : intercept = b

Slope = -2: intercept = 4

d Horizontal intercept = a

(4) 4 2 Horizontal intercept = 2 2

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

21

Plot the line 4x+2y - 8 = 0 by calculating the horizontal and vertical intercepts:
4x+2y

-8=0

y
6 5 4 3 2 1 0 -1 -1 -2 0 1 2

Rearrange the equation into the form y = mx + c vertical intercept 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

(0, 4)

y = - 2x + 4

(1, 2)

(2, 0)
3 4

(3, -2) horizontal intercept

Figure 2.13

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

22

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
y
6 5 4 3

y=4

x = - 1.5

2 1 0

x=2

-4

-3

-2

-1 -1 -2

y=-2

-3 -4

Figure 2.11 23

Copyright2001 Teresa Bradley and John Wiley & Sons Ltd