1 copyright©2001 teresa bradley and john wiley & sons ltd chapter 2: the straight line and...
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1Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
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
2Copyright©2001 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’
Slope =
-5
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
intercept = 2
intercept = 0
intercept = - 3
change in height
change in distance
y
x
6
52
4
Line CD
slope =
Line AB
slope = Figure 2.6
-6
-5
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
3Copyright©2001 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
Lines with same slope but different intercepts are different lines
-5
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
-2
-1
0
1
2
3
4
5
6
-1 0 1 2 3 4 5 6
4Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
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
0
1
2
3
4
5
-1 0 1 2 3 4
Intercept, c = 2
slope, m = 1
5Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
Draw the line, given slope =1: intercept = 2Worked 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
1
2
3
4
5
-1 0 1 2 3 4
(0, 2)
( 1, 3)
x
Figure 2.6
6Copyright©2001 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 5 6
y 0 1 2 3 4 5 6 Plot the points as follows
7Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
x 0 1 2 3 4 5 6y 0 1 2 3 4 5 6
0
1
2
3
4
5
6
0 1 2 3 4 5 6x
y
(0, 0)
Plot the point x = 0, y = 0
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x 0 1 2 3 4 5 6y 0 1 2 3 4 5 6
0
1
2
3
4
5
6
0 1 2 3 4 5 6x
y
(0, 0)
(1, 1)
Plot the point x = 1, y = 1
9Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
x 0 1 2 3 4 5 6y 0 1 2 3 4 5 6
0
1
2
3
4
5
6
0 1 2 3 4 5 6x
y
(2, 2)
(1, 1)
(0, 0)
Plot the point x = 2, y = 2
10Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
x 0 1 2 3 4 5 6y 0 1 2 3 4 5 6
0
1
2
3
4
5
6
0 1 2 3 4 5 6x
y
(2, 2)
(1, 1)
(0, 0)
(3, 3)
Plot the point x = 3, y = 3
11Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
x 0 1 2 3 4 5 6y 0 1 2 3 4 5 6
0
1
2
3
4
5
6
0 1 2 3 4 5 6x
y
(2, 2)
(1, 1)
(0, 0)
(3, 3)
(4, 4)
Plot the point x = 4, y = 4
12Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
x 0 1 2 3 4 5 6y 0 1 2 3 4 5 6
0
1
2
3
4
5
6
0 1 2 3 4 5 6x
y
(2, 2)
(1, 1)
(0, 0)
(3, 3)
(4, 4)
(5, 5)
Plot the point x = 5, y = 5
13Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
x 0 1 2 3 4 5 6y 0 1 2 3 4 5 6
0
1
2
3
4
5
6
0 1 2 3 4 5 6x
y
(2, 2)
(1, 1)
(0, 0)
(3, 3)
(4, 4)
(5, 5)
(6, 6)
Plot the point x = 6, y = 6
14Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
x 0 1 2 3 4 5 6y 0 1 2 3 4 5 6
0
1
2
3
4
5
6
0 1 2 3 4 5 6x
y
(2, 2)
(1, 1)
(0, 0)
(3, 3)
(4, 4)
(5, 5)
(6, 6)
Join the plotted points
15Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
x 0 1 2 3 4 5 6y 0 1 2 3 4 5 6
0
1
2
3
4
5
6
0 1 2 3 4 5 6x
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 45o line, through the origin
16Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
x 0 1 2 3 4 5 6y 0 1 2 3 4 5 6
0
1
2
3
4
5
6
0 1 2 3 4 5 6x
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
17Copyright©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
1
2
3
4
5
-1 0 1 2 3 4
(0, 2)
( 1, 3)
x
y
(2, 4)
Figure 2.6
18Copyright©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
1
2
3
4
5
-1 0 1 2 3 4
(0, 2)
( 1, 3)
x
y
(2, 4)
Figure 2.6
19Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
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 for any point on the line, when given the value of the x co-ordinate.
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
The equation of a line may be written in terms of the two
characteristics, m (slope) and c (intercept) . y = mx + c
20Copyright©2001 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 + cIntercept = c
Slope = m
Horizontal intercept = - c/m
0, 0
21Copyright©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
Rearrange the equation into
the form y = mx + c:
Slope = : intercept =
Horizontal intercept =
Example:4x + 2y - 8 = 0
Slope = -2: intercept = 4
Horizontal intercept =
ax by d
by d ax
yd
b
a
bx
0
a
b
d
b
d
a
4 2 8 0
2 8 4
4 2
x y
y x
y x
( )4
2
4
22
22Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
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
(see previous slide) Plot these points: see Figure
2.13 Draw the line thro’ the points Figure 2.13
-2
-1
0
1
2
3
4
5
6
-1 0 1 2 3 4
vertical intercept
horizontal intercept
y
(3, -2)
x
y = - 2x + 4
(2, 0)
(0, 4)
(1, 2)
23Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
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
-4
-3
-2
-1
0
1
2
3
4
5
6
-4 -3 -2 -1 0 1 2 3 4
y = 4
x = 2 x = - 1.5
y = - 2
x
y
Figure 2.11