OxfordAQA International GCSE Mathematics: Graphs in real contexts, rates of change and variation (9260) – Revision Notes

These revision notes condense references A17 and A18 (section 3.2.2, Functions, graphs and calculus) of the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1 (for exams May/June 2018 onwards). Full explanations and longer worked examples are in the study guide. Test yourself with the practice set. See the OxfordAQA International GCSE Mathematics course hub, the printable checklist tracks progress, and the free diagnostics show where to focus.

A17 has Core and Extension content. A18 is Extension only, as are gradients and areas of graphs (including areas under velocity-time graphs and curved graphs).

Key results at a glance

Idea	Rule
Gradient	Change up ÷ change across
As a rate	Units are y-unit per x-unit
Vertical intercept	Value of y when x = 0 (the starting amount)
Speed	distance ÷ time
Acceleration	change in speed ÷ time
m/s to km/h	× 3.6
km/h to m/s	÷ 3.6
Area under a velocity-time graph (Extension only)	Distance travelled
Gradient of a distance-time graph	Speed
Gradient of a velocity-time graph	Acceleration

Reading real-context graphs

Method in steps:

1. Read the axis labels and units before anything else.
2. To find y from x, go up from x to the graph, then across.
3. To solve “when does the quantity equal c?”, draw a horizontal line at c and read the x values where it meets the graph.
4. Where two graphs cross, the two quantities are equal at that point.
5. Give units and write “approximately” or “about”.

A graph with a steep part and a shallow part shows the rate changing. A horizontal part means no change in that quantity, which in a distance-time graph means standing still.

Worked reminder

A water bill is a fixed charge plus a cost per cubic metre. The line passes through (0, 18) and (20, 66), with cost in pounds.

Gradient = (66 - 18)/(20 - 0) = 48/20 = 2.4

The gradient means £2.40 per cubic metre. The intercept means a £18 fixed charge. The cost for 35 cubic metres is 18 + 2.4 × 35 = £102.

Comparing two graphs

A phone plan costs £8 plus £0.50 for each gigabyte. A second plan costs £3 plus £0.80 for each gigabyte.

8 + 0.5g = 3 + 0.8g
5 = 0.3g
g = 16.67 (to 2 d.p.)

The lines cross at about 17 gigabytes. Below that the second plan is cheaper, because it starts lower. Above it the first plan is cheaper, because its gradient is smaller.

Straight-line gradient as rate of change

• A positive gradient: quantity increasing. A negative gradient: decreasing.
• A larger gradient (in size) means a faster rate.
• Always state the meaning in context: “The tank loses 8 litres every minute.”
• A straight line means the rate is constant. A curve means the rate is changing.

Kinematics

Distance-time graphs

Feature	Meaning
Straight upward line	Steady movement away from the start
Flat line	Not moving
Straight downward line	Constant speed back towards the start
Steeper line	Faster

Average speed = total distance travelled ÷ total time. Never use displacement for this.

Velocity-time graphs

Feature	Meaning
Horizontal line	Constant speed, no acceleration
Upward slope	Speeding up
Downward slope	Slowing down (negative acceleration)
Area under the line (Extension only)	Distance travelled

(Extension only) For a straight-line graph, find the area as a rectangle, triangle or trapezium: area = ½ × (sum of the two parallel sides) × height.

Worked reminder (acceleration Core; distance Extension only)

A lift goes from rest to 3 m/s in 4 s, moves at 3 m/s for 10 s, then stops in 2 s.

Acceleration = 3/4 = 0.75 m/s²
Deceleration = 3/2 = 1.5 m/s²
Distance = ½ × (10 + 16) × 3 = 39 m

The parallel sides are 10 s (constant speed) and 16 s (whole journey), and the height is 3 m/s.

Unit changes

54 km/h ÷ 3.6 = 15 m/s
8 m/s × 3.6 = 28.8 km/h

Be consistent: if speed is in km/h, convert minutes to hours first (20 min = 1/3 h).

Curved graphs (Extension only)

Task	Method
Gradient at a point	Draw a tangent with a ruler, then use two far-apart points on the tangent
Area under a curve	Split into equal strips and use trapezia

Trapezium estimate for equal strips of width h:

Area ≈ (h/2) × [first + last + 2 × (sum of the middle readings)]

Whether the estimate is high or low depends on the curve’s shape. Where the curve bends downwards, trapezia tend to give an underestimate. Where it bends upwards, they tend to give an overestimate.

Worked reminder (Extension only): tangent

On a velocity-time graph, a tangent at t = 6 s passes through (2, 5) and (10, 11) m/s.

Gradient = (11 - 5)/(10 - 2) = 6/8 = 0.75

The acceleration at t = 6 is about 0.75 m/s².

Worked reminder (Extension only): trapezia

Speed readings every 5 s are 0, 4, 6, 7 and 8 m/s from t = 0 to t = 20.

Area ≈ (5/2) × [0 + 8 + 2 × (4 + 6 + 7)]
     = 2.5 × [8 + 34]
     = 105 m

Direct and inverse variation (Extension only)

Words	Equation
y ∝ x	y = kx
y ∝ x²	y = kx²
y ∝ x³	y = kx³
y ∝ √x	y = k√x
y inversely ∝ x	y = k/x
y inversely ∝ x²	y = k/x²
y inversely ∝ √x	y = k/√x

Method in steps:

1. Write the equation with k.
2. Substitute the known pair to find k.
3. Write the formula with that value of k.
4. Substitute the new value and solve.

Worked reminder

y is proportional to x³, and y = 40 when x = 2. Find y when x = 3 and x when y = 320.

y = kx³, 40 = 8k, k = 5
x = 3: y = 5 × 27 = 135
y = 320: x³ = 64, x = 4

Must-know distinctions

Direct	Inverse
Both quantities rise together	One rises as the other falls
y ÷ xⁿ stays fixed	y × xⁿ stays fixed
Graph starts at the origin	Graph never meets either axis
Double x: y doubles (y = kx)	Double x: y halves (y = k/x)

For y = kx², doubling x multiplies y by 4. For y = k/x², doubling x divides y by 4.

Quick self-test

1. A straight line passes through (2, 5) and (10, 29). Find its gradient.
2. A tank fills at a constant rate. After 8 minutes it holds 56 litres, starting empty. What is the rate in litres per minute?
3. A walker covers 5 km in 15 minutes at steady speed. Give the speed in km/h.
4. Convert 90 km/h to m/s.
5. A scooter’s speed goes from 4 m/s to 16 m/s in 6 s. Find its acceleration.
6. (Extension only) A bus starts from rest and reaches 6 m/s after 5 s, with uniform acceleration. How far does it travel in this time?
7. (Extension only) y is directly proportional to x. y = 21 when x = 7. Find y when x = 12.
8. (Extension only) y is inversely proportional to x. y = 5 when x = 8. Find y when x = 2.
9. (Extension only) y is proportional to x³ and y = 16 when x = 2. Find x when y = 54.
10. (Extension only) y is inversely proportional to √x and y = 4 when x = 9. Find x when y = 3.

Answers

1. (29 - 5)/(10 - 2) = 24/8 = 3
2. 56 ÷ 8 = 7 litres per minute
3. 15 minutes = 0.25 h, so 5 ÷ 0.25 = 20 km/h
4. 90 ÷ 3.6 = 25 m/s
5. (16 - 4) ÷ 6 = 2 m/s²
6. Area of a triangle = ½ × 5 × 6 = 15 m
7. y = kx, k = 21/7 = 3, y = 3 × 12 = 36
8. k = 5 × 8 = 40, y = 40/2 = 20
9. y = kx³, 16 = 8k, k = 2; 54 = 2x³, x³ = 27, x = 3
10. y = k/√x, 4 = k/3, k = 12; 3 = 12/√x, √x = 4, x = 16

Where marks are usually lost

• Giving a gradient with no units, or with units that do not match the axes.
• Quoting the gradient as a plain number when the question asks what it represents.
• Drawing a line that cuts the curve instead of touching it when estimating a gradient (Extension only).
• Finding average speed from the distance at the end instead of the total distance travelled.
• Mixing minutes and hours, or metres and kilometres, in one calculation.
• Using the whole time as a parallel side of a trapezium when only the constant-speed part is the other side.
• Substituting the new value into the formula before finding k (Extension only).
• Treating “inversely proportional” as y = kx, or “proportional to x²” as y = kx.
• Leaving a read-off answer without saying it is approximate, or giving a value far outside a sensible reading range.

Official syllabus

OxfordAQA International GCSE Mathematics (9260), specification Version 5.1 (for exams May/June 2018 onwards), published by OxfordAQA, section 3.2.2 Functions, graphs and calculus, references A17 and A18.