EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Real-life graphs
Graphs · Lesson 4 of 8
Last Lesson and Before
Answer each one, then check.
1. Last lesson: what does the gradient of a distance-time graph show?
Speed
2. What is the y-intercept of y = 1.5x + 3?
3
3. 5 miles is about 8 km. About how many km is 20 miles?
32 km
4. Work out 20 + 0.05 × 300.
35
Learning Objectives
1. Use conversion graphs.
2. Draw and read cost graphs, and explain the gradient and intercept.
3. Use graphs to compare two options.
4. Match container shapes to graphs of depth against time.
A Conversion Graph
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A conversion graph is a straight line through the origin, because the two units are in direct proportion. Read up from one axis to the line, then across to the other. For values off the graph, scale up: 125 miles is 5 × 25 miles, so about 5 × 40 = 200 km. |
25 miles is about 40 km. |
Cost Graphs
A taxi charges a fixed £3, plus £1.50 per mile: C = 1.5m + 3.
▸ The intercept. Where the line meets the cost axis: the fixed charge, paid even for 0 miles.
▸ The gradient. The cost per mile: £1.50.
▸ Not through the origin. A fixed charge means cost is NOT directly proportional to distance.
▸ Reading off. 8 miles: 1.5 × 8 + 3 = £15.
Comparing Two Deals
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Phone plan A costs £20 a month plus 5p per minute. Plan B costs 15p per minute, with no monthly charge. For how many minutes do they cost the same? Which is cheaper for 300 minutes? |
1. Write a formula for each
A: C = 20 + 0.05m; B: C = 0.15m
2. They cost the same where the lines cross
20 + 0.05m = 0.15m
3. Solve
20 = 0.1m, so m = 200
4. 300 minutes
A: 20 + 15 = £35; B: £45
Answer: The same at 200 minutes; plan A is cheaper for 300 minutes.
Filling Containers
Water pours in at a steady rate. The graph of depth against time depends on the width of the container.
▸ Straight sides. A cylinder or cuboid fills at a steady rate: a straight line.
▸ Wider at the top. The depth rises quickly at first, then more slowly: a curve that gets less steep.
▸ Narrower at the top. The depth rises slowly at first, then faster: a curve that gets steeper.
▸ Wide then narrow parts. Each section of the container gives its own section of graph; a narrow section gives a steeper part.
Shape and Graph
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Container |
How the width changes |
Graph of depth |
|---|---|---|
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Cylinder |
Same width all the way up |
Straight line |
|
Vase, wide at the top |
Gets wider |
Curve, getting less steep |
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Cone, point up |
Gets narrower |
Curve, getting steeper |
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Flask: wide base, narrow neck |
Wide, then narrow |
Line, then a steeper line |
Key Terms
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Conversion graph A graph for changing between two units. |
Fixed charge A cost that is paid whatever the amount used; the y-intercept. |
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Direct proportion Two quantities whose graph is a straight line through the origin. |
Break-even point Where two cost lines cross, so both options cost the same. |
Your Task: Which Taxi?
12 minutes
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Taxi X charges £2 plus £2 per mile. Taxi Y charges £6 plus £1.50 per mile. Draw both graphs for 0 to 12 miles on the same axes. For which journeys is each taxi cheaper? 1. Make a table of values for each taxi. 2. Plot both lines on the same axes. 3. Find where they cross. |
A good answer shows: X: C = 2m + 2; Y: C = 1.5m + 6. They cross where 2m + 2 = 1.5m + 6, so m = 8 miles (£18). X is cheaper under 8 miles; Y is cheaper over 8 miles.
Can I...?
☐ Read a conversion graph both ways.
☐ Use a conversion graph for values off the scale.
☐ Explain the gradient of a real-life graph.
☐ Explain the intercept of a real-life graph.
☐ Compare two options using graphs.
☐ Match a container to its depth-time graph.
Summary
✓ Conversion graph: straight line through the origin.
✓ Cost graph: intercept = fixed charge; gradient = cost per unit.
✓ Two lines cross where the options cost the same.
✓ Narrow parts of a container fill faster: a steeper graph.
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EXAM FOCUS Plan A costs £20 a month plus 5p per minute. Plan B costs 15p per minute. Jo uses about 300 minutes a month. Which plan should she choose? (3 marks) When asked what the gradient or intercept "represents", answer in context with units: "the cost per mile is £1.50", not just "the gradient is 1.5". |