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• ISBN 9780170351027 Chapter 14 Graphing lines 285

14GraphinG lines

In thIs Chapter you will:

use a linear equation to complete a table of values

identify points and quadrants on a number plane

graph tables of values on the number plane

graph linear equations on the number plane

test if a point lies on a line

find the equation of horizontal and vertical lines

solve linear equations graphically

what’s in Chapter 14? 14–01 tables of values 14–02 the number plane 14–03 Graphing tables of values 14–04 Graphing linear equations 14–05 testing if a point lies on a line 14–06 horizontal and vertical lines 14–07 solving linear equations graphically

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• Developmental Mathematics Book 3 ISBN 9780170351027 286

14–01 tables of values

example 1

Complete each table of values using the equation given.

a y = x + 2 b y = 2x – 1

x 0 1 2 3

y

x −1 1 4

y

c d = 3c + 5

c −2 1 4

d

solutIon

Substitute the x-values from the table into each equation.

a y = x + 2 When x = 0, y = 0 + 2 = 2 When x = 1, y = 1 + 2 = 3, and so on.

x 0 1 2 3

y 2 3 4 5

b y = 2x – 1 When x = –1, y = 2 × (−1) − 1 = −3 When x = 1, y = 2 × 1 − 1 = 1, and so on.

x −1 1 4

y −3 1 7

c d = 3c + 5 When c = –2, d = 3 × (−2) + 5 = −1 When c = 1, d = 3 × 1 + 5 = 8, and so on.

c −2 1 4

d −1 8 17 Shu tt

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av on

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• 287 ISBN 9780170351027 Chapter 14 Graphing lines

1 Find the value of y when x = 2 if y = 2x – 3. Select the correct answer A, B, C or D.

A –7 B –1 C 1 D –5

2 Find the value of m when n = –3 if m = 8 – 2n. Select A, B, C or D.

A 2 B 14 C 6 D 3

3 Copy and complete each table of values. a y = 2x

x 0 1 2 3

y

b y = x ÷ 2

x 10 8 6 4

y

c y = x – 1

x 4 3 2 1

y

d y = x + 3

x 0 1 2 3 4

y

e y = 3x

x 1 2 3 4

y

f y = x – 3

x 7 6 5 4 0 −1

y

g y = 2 + x

x 0 1 2 3 4

y

h y = x 3

x 12 9 6 3 0 −3 −6

y

i y = 2x + 1

x 1 2 3 4

y

j y = 3x – 1

x 4 3 2 1 0 −1 −2 −3

y

4 Copy and complete each table of values.

a d = 5c + 1

c −1 0 1

d

b h = 2g – 3

g 1 2 3

h

c q = 6 + 2p

p −1 0 1

q

d t = 12 – 4s

s 1 2 3

t

e n = 3 – 2m

m −2 0 2

n

f y = 5x – 6

x 1 3 5

y

exercise 14–01

• Developmental Mathematics Book 3 ISBN 9780170351027 288

the number plane14–02 A number plane is a grid for plotting points and drawing graphs. It has an x-axis which is horizontal (goes across) and a y-axis which is vertical (goes up and down). The origin is the centre of the number plane. The number plane is divided into 4 quadrants (quarters).

example 2

Plot each point on a number plane.

A(1, 2), B(–2, 3), C(–3, –4), D(2, –4), E(0, 3), F(–4, 0), G(2, 0), H(0, –1).

solutIon

y

E B

F

A

x G

D

H

C

4

3

2

1

–1 1 2 3 4–4 –3 –2 –1

–2

–3

–4

A(1, 2) is 1 unit right and 2 units up from the origin.

C(–3, –4) is 3 units left and 4 units down from the origin.

example 3

In which quadrant does each point lie?

A (–1, 4) B ( 2, –3) C (–3, –3) D (0, 2)

solutIon

A is in the 2nd quadrant.

B is in the 4th quadrant.

C is in the 3rd quadrant.

D is on the y-axis, so it is not in any quadrant (between the 1st and 2nd quadrants).

y

x

the origin (0, 0)

iS to

ck ph

ot o/

G iz

m o

y A

D

x

BC

4

3

2

1

–1 1 2 3 4–3 –2 –1

–2

–3

–4

• ISBN 9780170351027 Chapter 14 Graphing lines 289

1 In which quadrant does the point (–2, –4) lie?

2 Where is the point (0, 0) positioned on the number plane? Select A, B, C or D.

A Below the x-axis B Below the y-axis

C Where the x and y-axes meet D In the 1st quadrant

3 a Write the coordinates of each point A to F shown.

b State which quadrant or axis each point lies in.

4 a On a number plane, plot the points below.

a A(1, 2) b B(–l, 2) c C(1, –2) d D(–l, –2) e E(3, –5)

f F(–5, –2) g G(4, –3) h H(0, 4) i I(2, 0) j J(–2, –3)

k K(0, –3) l L(–3, 0) m M(–4, –1) n N(2, –5) o P(4, 5)

p Q(3, 3) q R(2, 2) r S(–2, –2) s T(0, 0) t V(0, –5)

b What type of figure is formed by the points:

i ABCD? ii LFS? iii ATBH?

5 Picture puzzle: Draw a number plane with the x-axis from −10 to 10 and the y-axis from −4 to 6. Plot the points described below and join them as you go. Do not join points separated by a line. What familiar shape is formed?

(10, 4) (−10, 4) (−8, 0) (−1, 0) (−4, 3) (8, 4) (−8, 4) (0, 0) (−1, 3) (−4, 6) (8, −2) (−8, −2) (8, 0) (−3, 3) (10, −2) (−10, −2) (−2, 0) (−2, 6) (10, 4) (−10, 4) (6, 0) (−2, 3) (−1, 3)

(6, 1) (0, 6) (8, 2) (−8, −1) (−3, 0) (1, 3) (7, 3) (−7, 0) (5, 0) (−3. 3) (2, 6) (6, 4) (−6, 1) (5, 2) (3, 3) (5, 5) (−5, 2) (−4, 0) (4, 6) (4, 6) (−4, 3) (4, 0) (−4, 3) (4, 3) (3, 6) (−3, 3) (4, 3) (6, 4) (2, 6) (−2, 3) (−5, 0) (5, 2) (1, 6) (−1, 3) (3, 0) (−5, 2) (7, 3) (0, 6) (0, 3) (3, 3) (6, 1) (−1, 6) (1, 3) (−6, 0) (8, 2) (−2, 6) (2, 3) (2, 0) (−6, 1) (7, 0) (−3, 6) (3, 3) (2, 3) (−4, 6) (4, 3) (−7, 0) (−5, 5) (5, 2) (1, 0) (−8, 2) (−6, 4) (6, 1) (1, 3) (−6, 1) (−7, 3) (7, 0) (−7, 3) (−8, 2) (8, −1) (0, 0) (−5, 2)

(0, 3) (−6, 4)

exercise 14–02

y D

E

x

A

C

B

F

3

2

1

–1 1 2 3 4 5 6–4–5 –3 –2 –1

–2

–3

• Developmental Mathematics Book 3 ISBN 9780170351027 290

Graphing tables of values14–03

example 4

x −2 −1 0 1 2

y −1 0 1 2 3

Graph this table of values on a number plane.

solutIon

Reading the table of values in columns, we get the coordinates of the points.

x −2 −1 0 1 2

y −1 0 1 2 3

The points are: (−2, −1) (−1, 0) (0, 1) (1, 2) (2, 3)

Plotting these points on a number plane:

y

x

3

2

1

–1 1 2 3–3 –2 –1

–2

–3

example 5

Graph this table of values after completing it.

y = 2x – 3

x −1 2 0 3

y

solutIon

x −1 2 0 3

y −5 1 −3 3

y

x

3

2

1

–1 1 2 3–3 –2 –1

–2

–3

–4

–5

• ISBN 9780170351027 Chapter 14 Graphing lines 291

1 If y = 4 – x, find y when x = –2. Select the correct answer A, B, C or D.

A 2 B 6 C –2 D –6

2 Graph each table of values on a number plane. a

x −2 −1 0 1 2

y 0 1 2 3 4

b x −2 −1 0 1 2

y −4 −2 0 2 4

c x −2 −1 0 1 2

y −5 −4 −3 −2 −1

d x −2 −1 0 1 2

y −1 − 1 2

0 1 2

1

e x −2 −1 0 1 2

y 4 2 0 −2 −4

f x −3 −1 0 2 3

y −5 −1 1 5 7

g x −4 −2 0 1 2

y 1 3 5 6 7

h x −2 −1 0 2 3

y −6 −5 −4 −3 −2

i x −2 −1 0 1 2

y −7 −4 −1 2 5

j x −3 −1 0 2 3

y 7 3 1 −3 −5

3 What do you notice about each set of points in question 2?

4 Copy and complete each table and then graph the values on a number plane. a y = x + 3

x −1 0 1 2

y

b y = 6 – x

x −1 0 1 2

y

c y = 2x + 1

x −2 0 1 2

y

d y = 4x – 3

x −1 1 0 2

y

e y = –3 + x

x −1 0 1 2

y

f y = 12 – 5x

x 1 0 3 2

y

exercise 14–03

• Developmental Mathematics Book 3 ISBN 9780170351027 292

Graphing linear equations14–04

To be sure, it is best to find three points on the line using a table of values. We can substitute any x-values into the linear equation, but x = 0, x = 1 or x = 2 are usually the easiest to use.

example 6

Graph each linear equation and state the x-intercept and y-intercept of each line.

a y = 3x b y = 2x –1

solutIon

a Complete a table of values. y = 3x

x 0 1 2

y 0 3 6

Graph the table of values, rule the line and label it with the equation.

Draw arrows on the ends of the line because a line has an infinite number of points and goes on endlessly in both directions.

The line crosses the x-axis at 0, so its x-intercept is 0.

The line crosses the y-axis at 0, so its y-intercept is also 0.

b y = 2x –1

x 0 1 2

y −1 1 3

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