chap2maths

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1Copyright2001 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 interceptSlides 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

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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

5

2

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

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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

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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

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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 horizontallyforward 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

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What does the equation of a line mean

Consider the equationy =x

From the equation, calculate the values ofy 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

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7/237Copyright2001 Teresa Bradley and John Wiley & Sons Ltd

x 0 1 2 3 4 5 6

y 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 pointx = 0,y = 0

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x 0 1 2 3 4 5 6

y 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)


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x 0 1 2 3 4 5 6

y 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)


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x 0 1 2 3 4 5 6

y 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)


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x 0 1 2 3 4 5 6

y 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)


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x 0 1 2 3 4 5 6

y 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)


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x 0 1 2 3 4 5 6

y 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)


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x 0 1 2 3 4 5 6

y 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

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x 0 1 2 3 4 5 6

y 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)

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

Figure 2.9 The 45o line, through the origin

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x 0 1 2 3 4 5 6

y 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

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Deduce the equation of the line, given

slope,m = 1; intercept,c = 2

1. Determine and plot at least 2 points:

2. Start atx = 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 they co-ordinate is

always (value of thex 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

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Deduce the equation of the line, given

slope,m = 1; intercept,c = 2

Use Formulay = mx +c

Since m = 1, c = 2 , then

y = mx + c y = 1x + 2

y = x + 2See 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

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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 they co-ordinatefor 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 whichhas 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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Calculating the Horizontal Intercepts

Calculate the horizontal intercept

for the line:y =mx +c

The horizontal intercept is the point

where the line crosses thex -axis

Use the fact that they co-ordinate is

zero at every point on thex-axis.

Therefore, substitute y = 0 into the

equation of the line

0 = mx + c

and solve forx: 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

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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 formy = 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

( )42

42

2

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Plot the line 4x+2y - 8 = 0 by calculating the

horizontal and vertical intercepts:

4x+2y - 8 = 0

Rearrange the equation into

the formy = mx + c

y = -2x + 4 Vertical intercept aty = 4

Horizontal intercept atx = 2

(see previous slide)

Plot these points: see Figure2.13

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

(3, -2)

x

y = - 2x + 4

(2, 0)

(0, 4)

(1, 2)

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Equations of Horizontal and vertical lines:

The equation of a horizontal is given by the point of intersection with

they-axis

The equation of a vertical line is given by the point of intersection with

thex -axis

-4

-3

-2

-1

0

1

2

3

4

5

6

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

y = 4

x = 2x = - 1.5

y = - 2

x

y

Figure 2.11

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