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9.02 Linear graphs

Worksheet
Identify key features
1

For each of the following linear equations:

i

State the gradient of the line.

ii

State the y-intercept of the line.

a
y = 4x + 11
b
y = \dfrac{3x}{2}- 10
c
y = - 1 - \dfrac{9 x}{2}
d
y = \dfrac{9 - 2 x}{4}
2

Which of the following linear relationships represents a more rapid change in the value of y?

A
y+5x=3
B
x012
y369
3

Which of the following linear relationships has a larger x-intercept?

A
y=2x-2
B
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
4

Consider the straight line shown in the following graph:

Does the line y = 6 x + 1 have a larger or smaller rate of change than the graphed line?

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
5

Consider the straight line shown in the following graph:

Are the y-values of the line y = 6 x + 2 changing more or less rapidly than in the graphed line?

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

Consider the straight line shown in the following graph:

Does the line y = - 4 x + 6 have a larger or smaller y-intercept than the graphed line?

-4
-3
-2
-1
1
2
3
4
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
Find the equation of a line
7

Find the equation of the following lines in gradient-intercept form:

a

A line that has a gradient of \dfrac{4}{3} and passes through the point \left(0, - 3 \right)

b

A line that has a gradient of \dfrac{2}{3} and passes through the point \left(0,3 \right)

c

A line that passes through the points A \left( - 6 , - 5 \right) and B \left(1, 6\right)

8

Find the equation of a line that has the same gradient as the line y = 7 - 3 x and the same

y-intercept as the line y = - 7 x - 8.

9

A line has gradient 5 and passes through the point \left( - 1 , - \dfrac{10}{3} \right). Find the equation of the line in general form.

10

A line has a gradient of - 2 and passes through the point \left( - 6 , - 3 \right). Find the equation of the line in gradient-intercept form.

11

A straight line passes through the point \left(0, \dfrac{3}{4} \right) with gradient 2.

a

Find the equation of the line in gradient-intercept form.

b

Find the equation of the line in general form.

c

Find the x-intercept of the line.

12

Consider the graph of the line:

a

What is the value of the y-intercept?

b

What is the gradient of the line?

c

Find the equation of the line in gradient-intercept form.

d

Rewrite the equation of the line in general form.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
5
6
7
y
13

For the following graphed lines:

i

State the gradient.

ii

State the y-intercept.

iii

Write the equation of the line in gradient-intercept form.

a
1
2
3
4
5
6
7
8
9
10
x
-4
-3
-2
-1
1
2
3
4
y
b
-7
-6
-5
-4
-3
-2
-1
1
x
-4
-3
-2
-1
1
2
3
4
y
14

Consider the line with equation 2 x + y - 8 = 0.

a

Find the x-intercept of the line.

b

Find the equation of a line with a gradient of - 4 that passes through the x-intercept of the given line.

15

For each of the following pairs of points:

i

Find the gradient of the line that passes through both points.

ii

Find the equation of the line in gradient-intercept form.

a

\left(0, 6\right) and \left(3, 18\right)

b

\left( - 2 , - 9 \right) and \left( - 5 , 12\right)

c

\left( - 2 , 2\right) and \left( - 5 , 7\right)

d

\left(0, - 3 \right) and \left(1, 4\right)

16

Consider the equation of the line 5 x - 4 y + 20 = 0.

a

Find the x-intercept of the line.

b

Hence, find the equation of the line that passes through \left(2, 8\right) and the x-intercept. Write your answer in gradient-intercept form.

17

Consider the table of values below:

a

Is y increasing or decreasing?

b

For every 1 unit increase in x, by how much does y change?

c

Hence, find the algebraic rule linking x and y.

x9182736
y-68-131-194-257
18

Write an equation for y in terms of x for the values in the given table:

x0...891011
y-6...18212427
19

A line has gradient - 2 and passes through the point \left( - 6 , - \dfrac{4}{3} \right). Find the equation of the line in general form.

20

For each of the following equations:

i

Rewrite the equation in gradient-intercept form.

ii

Find the gradient of the line.

iii

Find the y-intercept of the line.

a

y = 3 \left( - 4 x - 2\right)

b

y = 6 \left( 3 x - 2\right)

21

A line has the equation 3 x - y - 4 = 0.

a

Write the equation of the line in gradient-intercept form.

b

Find the gradient of the line.

c

Find the y-intercept of the line.

22

Find the equation of the vertical line that passes through the point \left(2, 3\right).

23

The rectangle PQRS has vertices at P\left(-4,-3\right), Q\left(-6,-3\right), R\left(-6,-5\right) and S\left(-4,-5\right).

a

Find the equation of the line PQ.

b

Is the line QR vertical or horizontal?

c

Find the equation of the line RS.

d

Is the line PS vertical or horizontal?

e

Find the equation of the line PS.

Sketch the graph of a linear relationship
24

Sketch the graph of the following linear equations by finding any two points on the line:

a

y = - 3 x - 4

b

y = \dfrac{x}{3} + 3

c
y = 5x-3
d
3x+y=6
e
2x-5y-10=0
f

- 6 x + 3 y + 24 = 0

25

Consider the equation y = - 4 x + 4.

a

Find the coordinates of:

i

The y-intercept

ii

The x-intercept

b

Sketch the graph for this equation.

26

Sketch the graph of the following linear equations using the gradient and y-intercept:

a

y = 4 x-1

b
y = -6x+2
c

y = \dfrac{1}{2} x - 2

d
2y = 3x-8
e
6x-3y=9
f
2x+5y=0
27

Sketch the graph of the following linear equations by finding the x-intercept and the y-intercept:

a
y=3x-9
b
5x+2y=10
c
6x-2y=18
d
4x-y+12=0
e

6 x + 2 y - 12 = 0

f

- 20 x + 5 y - 40 = 0

28

Sketch the graph of the following linear equations:

a
y = 4
b
x = 1
c
x=0
d
y=0
e
y = \dfrac{3}{2}
f
x = 4.5
g

y = - 3

h

x = - 6

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Outcomes

2.3.3

construct straight-line graphs both with and without the aid of technology

2.3.4

determine the slope and intercepts of a straight-line graph from both its equation and its plot

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