Gradient of a line Quiz (Grade 5)

Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8/9

Topic:

Score: 0 / 10

Gradient of a line — Notes & Worked Examples Show / Hide

Explanation:

The gradient (m) of a straight line measures how steep it is: Gradient = Change in yChange in x = y₂ − y₁x₂ − x₁.

Lines sloping upwards from left to right have a positive gradient; lines sloping downwards from left to right have a negative gradient; horizontal lines have gradient 0. Always subtract the coordinates in the same order on top and bottom.

Example 1: Find the gradient of the line passing through A(2, 3) and B(6, 15).

Solution:

Use m = y₂ − y₁x₂ − x₁:

m = 15 − 36 − 2 = 124 = 3

Answer: Gradient = 3

Example 2: Work out the gradient of the line joining P(−3, 7) and Q(5, 3).

Solution:

m = 3 − 75 − (−3) = −48

Simplify the fraction: m = −12 (or −0.5)

Answer: Gradient = −12

One Page Guide (PDF) Gradient of a line Worksheets

Question 1

[1 mark]

Work out
A line has a gradient of 2 and a y-intercept of 4
Write down the equation of the line

y =

Worked Solution & Answer

y = mx + c, where m = change in ychange in x is the gradient and c is the y-intercept (where x = 0).
y = 2x + 4

Answer: y = y = 2x + 4

Question 2

[1 mark]

Find the equation of the line that passes through (0, -6) and has a gradient of 5

y =

Worked Solution & Answer

y = mx + c, where m = change in ychange in x is the gradient and c is the y-intercept (where x = 0).
y = 5x - 6

Answer: y = y = 5x - 6

Question 3 AQA Jun 2018 2H Q12

[2 marks]

A straight line is drawn on the centimetre grid.
Fay assumes that the scale is 1 cm represents 1 unit.

(a) Use her assumption to work out the gradient of the line. [1 mark]

Answer

(b) In fact, the scale is 1 cm represents 2 units.
Which statement is correct?
Tick one box.
[ ] The answer to part (a) is too big
[ ] The answer to part (a) stays the same
[ ] The answer to part (a) is too small [1 mark]

Worked Solution & Answer

(a)

(Gradient = change in y ÷ change in x)
From (0, 1) to (6, 4): change in y = 3, change in x = 6
Gradient = 36 = 0.5 (or 12). B1

Answer (a): Answer 0.5

(b)

Both axes change by the same scale factor 2, so the ratio Δ yΔ x = 3 × 26 × 2 = 612 = 0.5 remains unchanged. B1

Answer (b): The answer to part (a) stays the same

Question Paper (PDF) • Mark Scheme (PDF)

Question 4 OCR Nov 2022 2F Q17

[3 marks]

A straight line, L, is shown below.

(a) Write down the ratio v : h when the gradient of line L is 4. [1 mark]

(b) Find the gradient of line L as a fraction in its simplest form when v : h = 14 : 6. [2 marks]

Worked Solution & Answer

(a)

Gradient = v / h = 4 ⇒ v : h = 4 : 1. B1

Answer (a): 4 : 1

(b)

(Gradient = change in y ÷ change in x)
Gradient = v / h = 14/6 = 7/3. M1 A1

Answer (b): 73

Question Paper (PDF) • Mark Scheme (PDF)

Question 5

[3 marks]

An electricity company charges the same fixed amount for each unit of electricity used.
David uses this graph to work out the total cost of the electricity he has used.

(a) Work out the gradient of the straight line. [2 marks]

(b) What does the gradient of this line represent? [1 mark]

Worked Solution & Answer

(a)

(Gradient = change in y ÷ change in x)
Line passes through (0, 0) and (100, 14) M1
⇒ gradient = 14 ÷ 100 = 0.14 A1

Answer (a): 0.14

(b)

Cost per unit of electricity

Answer (b): The cost per unit of electricity

Question Paper (PDF) • Mark Scheme (PDF)

Question 6

[3 marks]

The graph gives information about the volume, v litres, of petrol in the tank of Jim’s car after it has travelled a distance of d kilometres.

(a) Find the gradient of the graph. [2 marks]

(b) Interpret what the gradient of the graph represents. [1 mark]

Worked Solution & Answer

(a)

(Gradient = change in y ÷ change in x)
(0 – 27) ÷ (300 – 0) = –27 / 300 = –0.09 M1 A1

Answer (a): –0.09

(b)

The volume of petrol used (in litres) per kilometre travelled.

Answer (b): The volume of petrol used (in litres) per kilometre travelled.

Question Paper (PDF) • Mark Scheme (PDF)

Question 7 Practice Set 5 3F Q16

[4 marks]

(a) On the grid, draw the line x = 3 [1 mark]

(b) On this grid, draw the line y = x [1 mark]

Worked Solution & Answer

(a)

Vertical line at x = 3

Answer (a): Vertical line at x = 3

(b)

Diagonal line through (–4, –4) and (4, 4)

Answer (b): Diagonal line through (–4, –4) and (4, 4)

(c)

(Gradient = change in y ÷ change in x) M1
Gradient m = change in ychange in x = y₂ - y₁x₂ - x₁
Answer: 1.5 A1

Answer (c): 1.5

Question Paper (PDF) • Mark Scheme (PDF)

Question 8

[4 marks]

The graph shows the volume of liquid (L litres) in a container at time t seconds.

(a) Find the gradient of the graph. [2 marks]

(b) Explain what this gradient represents. [1 mark]

Worked Solution & Answer

(a)

(Gradient = change in y ÷ change in x)
Gradient = (16 – 4) ÷ (8 – 0) = 12 ÷ 8 = 1.5 M1 A1

Answer (a): 1.5

(b)

Rate of flow in litres per second C1

Answer (b): The rate at which the container fills (number of litres of liquid added per second)

(c)

Volume at t = 0 C1

Answer (c): The initial volume of liquid in the container at the start (when t = 0)

Question Paper (PDF) • Mark Scheme (PDF)

Question 9

[4 marks]

To cook rice
the number of cups of rice (x) : the number of cups of water (y) = 4 : 5

(a) Use this information to draw a graph to show the relationship between the number of cups of rice and the number of cups of water needed to cook rice. [2 marks]

(b) Find the gradient of the line drawn in part (a). [1 mark]

Worked Solution & Answer

(a)

Plot (0, 0), (4, 5) and (8, 10) and join M1 A1

Answer (a): Straight line drawn from (0, 0) through (4, 5) and (8, 10)

(b)

(Gradient = change in y ÷ change in x)
Gradient = 5 ÷ 4 = 1.25 B1

Answer (b): 1.25

(c)

Represents cups of water per 1 cup of rice C1

Answer (c): The number of cups of water needed per cup of rice (1.25 cups of water for every 1 cup of rice)

Question Paper (PDF) • Mark Scheme (PDF)

Question 10 Eduqas Nov 2018 1F Q19

[5 marks]

The graph shows the cost of delivery for goods bought at a furniture shop.

(a) Ceri buys a sofa from this shop and has it delivered.
Ceri lives 4 miles from the shop.
How much does Ceri pay for the delivery? [1 mark]

(b) Tick the correct statement about the distance and cost.
- They are not in proportion.
- They are in inverse proportion.
- They are in direct proportion.
- They are in indirect proportion. [1 mark]

Worked Solution & Answer

(a)

Reading from graph at distance = 4 miles gives cost = £18. B1

Answer (a): £ £18

(b)

The graph is a straight line through the origin (0, 0). B1

Answer (b): They are in direct proportion.

(c)

(Gradient = change in y ÷ change in x)
Gradient = 45 / 10 = 4.5. It represents the cost per mile (£4.50/mile). M2 A1

Answer (c): Gradient = 4.5; represents the cost per mile of delivery (£4.50 per mile).

Question Paper (PDF) • Mark Scheme (PDF)

Score: 0 / 10 (0 / 30 marks)

Gradient of a line Worksheets Next: Equation of a line →

Grade 5 Quizzes

25 Topics

5A - Combining ratios	5B - Direct and inverse proportion
5C - Reverse percentages	5D - Standard form
5E - Speed	5F - Changing the subject
5G - Quadratic expressions	5H - Solving quadratic equations
5J - Drawing quadratic graphs	5K - Cubic/reciprocal graphs
5L - Simultaneous equations	5M - Sector areas and arc lengths
5N - Sim. equ. graphically	5P - Midpoint of a line segment
5Q - Gradient of a line	5R - Equation of a line
5S - Spheres and cones	5T - Similar shapes (lengths)
5U - Trigonometry (SOHCAHTOA)	5V - Exact trig. values
5W - Probability trees	5X - Vectors
5Y - Venn diagrams	5Z - Density
5AA - Pressure