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Stage 5.1-3

4.02 The gradient of an interval

Worksheet
Gradient of a line
1

Describe the gradient of the lines in the following graphs:

a
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
y
b
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
c
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
d
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
2

For each of the following intervals between points A and B:

i

Find the rise.

ii

Find the run.

iii

Find the gradient.

a

A \left(2, 4\right) and B \left(8, 6\right)

1
2
3
4
5
6
7
8
9
x
1
2
3
4
5
6
7
8
9
y
b

A\left( - 3 , 4\right) and B\left(3, 16\right)

-4
-3
-2
-1
1
2
3
4
x
2
4
6
8
10
12
14
16
y
c

A \left( - 8 , 4\right) and B \left( - 1 , 18\right)

-9
-8
-7
-6
-5
-4
-3
-2
-1
1
x
2
4
6
8
10
12
14
16
18
y
d

A \left( - 5 , - 2 \right) and B \left(-1, 10\right)

-6
-5
-4
-3
-2
-1
1
2
3
4
x
-2
2
4
6
8
10
y
e

A \left( - 2 , - 1 \right) and B \left(2, - 13 \right)

-4
-3
-2
-1
1
2
3
4
x
-12
-10
-8
-6
-4
-2
2
y
f

A \left( - 1 , 2\right) and B \left(1, - 4 \right)

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
g

A \left(1, 3\right) and B \left(7, - 3 \right)

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

A \left( - 3 , - 2 \right) and B \left(5, 6\right)

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

Explain why a vertical line has an undefined gradient.

4

State the gradient of any line parallel to the x-axis.

5

Find the gradient of the following lines passing through points A and B:

a
-3
-2
-1
1
2
3
x
-3
-2
-1
1
2
3
y
b
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
c
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
x
-2
-1
1
2
3
4
5
6
y
d
-1
1
2
3
4
5
6
7
x
-1
1
2
3
4
y
Gradient formula
6

Find the gradient of the interval joining A \left( - 9 , 4\right) and B \left( - 3 , - 5 \right).

7

If we have two points and the slope formula m = \dfrac{y_2 - y_1}{x_2 - x_1}, does it matter which point is \left(x_1, y_1\right) and which point is \left(x_2, y_2\right)?

8

Consider the line plotted, where A \left(2, 0\right) and B \left(0, 4\right) both lie on the line.

a

Solve for the gradient of the line.

b

As x increases, what happens to the value of y?

-1
1
2
3
4
x
-1
1
2
3
4
y
9

Find the gradient of the line that passes through the given points:

a

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

b

\left( - 4 , 7\right) and \left(1, 10\right)

c

\left(1, - 4 \right) and the origin

d

\left(2, - 6 \right) and the origin

e

\left(6, 4\right) and \left(3, 4\right)

f

\left( - 6 , 5\right) and \left(4, 5\right)

g

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

h

\left( - 3 , - 1 \right) and \left( - 5 , 1\right)

10

Consider the points A \left( - 11 , - 9 \right), B \left( - 5 , 1\right) and C \left( - 2 , 6\right):

a

Find the gradient of AB.

b

Find the gradient of BC.

c

Do the points A, B and C lie in a straight line?

11

Consider the following points: A \left(26, m - 24\right), B \left( - 1 , m\right) and C \left( - 10 , 9\right).

Find m, given that A, B and C are collinear.

12

Given the gradient of the line passing through the two points, find the value of the pronumeral:

a

\left(4, - 3 \right) and \left(1, t\right), gradient = - 2

b

\left(5, 3\right) and \left(2, t\right), gradient = - 4

c

\left(5, 3\right) and \left(d, 63\right), gradient = 4

d

\left(11, c\right) and \left( - 20 , 16\right), gradient = - \dfrac{4}{7}

13

The line x = 4 intersects the line y = 2 x - 10 at the point Q.

The line x = - 4 intersects the line y = 2 x - 10 at point R.

a

Find the coordinates of Q.

b

Find the coordinates of R.

c

Find the gradient of a line that passes through Q and R.

14

Consider the line y = 5 x + 2 graphed:

a

Find the y-value of the point on the line where x = 5.

b

Find the gradient of the line.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
15

Consider the line y = - 2 x + 8.

a

Find:

i

y-intercept

ii

x-intercept

b

Sketch the line on a number plane.

c

Hence, find the gradient of the line.

16

Two lines L_{1} and L_{2} have equations \\y = x - 3 and y = - x + 5 respectively. The lines and their point of intersection have been graphed:

a

When x = 4, find the y-coordinate of the corresponding point on L_{1}.

b

When x = 4, find the y-coordinate of the corresponding point on L_{2}.

c

Find the gradient of the two lines:

i

L_{1}

ii

L_{2}

d

Find the product of their gradients.

-3
-2
-1
1
2
3
4
5
6
7
8
9
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
y
Properties of polygons
17

Consider the quadrilateral ABCD that has been graphed on the number plane:

a

Find the gradient of following sides:

i

AB

ii

CD

iii

AD

iv

BC

b

What type of quadrilateral is ABCD? Explain your answer.

-1
1
2
3
4
5
6
7
8
9
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
y
18

The 4 vertices of square ABCD have been plotted on a number plane.

a

Find the gradient of side AB.

b

Find the gradient of side BC.

c

Find the product of the gradients in parts (a) and (b).

d

If two lines are perpendicular their gradients multiply to -1. Are sides AB and BC perpendicular?

1
2
3
4
5
6
7
x
1
2
3
4
5
6
7
y
19

The points P \left(0, - 2 \right), Q \left( - 2 , 0\right), R \left(0, 4\right) and S \left(2, 2\right) are graphed below:

a
Find the gradient of PQ.
b

Find the gradient of RS.

c
Are PQ and RS parallel?
d
Are QR and PS parallel?
e
Identify the type of quadrilateral PQRS is.
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
20

The vertices of \triangle ABC are A \left(9, - 12 \right), B \left(4, 4\right) and C \left( - 8 , - 5 \right). The sides AB and AC have midpoints D and E respectively.

a

Find the coordinates of points D and E.

b

Find the gradient of side BC.

c

Find the gradient of side DE.

d

Are BC and DE parallel to each other?

21

A \left( - 4 , - 2 \right), B \left(2, 1\right) and C \left(2, - 4 \right) are the vertices of a triangle.

a

Name the side of the triangle that is a vertical line.

b

Find the area of the triangle.

Applications
22

Consider the following ramp:

a

Find the gradient of this skateboard ramp if it rises 0.9 metres above the ground and runs 1 metre horizontally at the base.

b

The ramp can only be used as a 'beginner’s ramp' if for every 1 metre horizontal run, it has a rise of at most 0.5 metres. Can it be used as a 'beginner’s ramp'?

23

A certain ski resort has two ski runs as shown in the diagram:

a

Find the gradient of Run A. Round your answer to two decimal places.

b

Find the gradient of ski run B. Round your answer to two decimal places.

c

Which run is steeper?

24

A paratrooper falls to the ground along a diagonal line. His fall begins 1157 \text{ m} above the ground, and the line he follows has a gradient of 1.3. That is, he falls 1.3 \text{ m} vertically for every 1 \text{ m} he moves across horizontally.

How far horizontally across the ground does he land from his initial position in the sky?

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Outcomes

MA5.3-8NA

uses formulas to find midpoint, gradient and distance on the Cartesian plane, and applies standard forms of the equation of a straight line

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