A10: Gradients & Intercepts

Foundation Higher AQAEdexcelOCREduqasCCEA

Identify and interpret gradients and intercepts of linear functions graphically and algebraically

Exam Board:

📋 Key Concepts

Gradient: The steepness of a line. Found by dividing vertical change by horizontal change.

Intercepts:

y-intercept: Where the line crosses the y-axis (when x = 0)
x-intercept: Where the line crosses the x-axis (when y = 0)

📝 Finding Gradient from a Graph

Method:

Choose two points on the line
Count the vertical change (up or down)
Count the horizontal change (right)
Divide: gradient = ^vertical⁄_horizontal

Example 1

A line passes through (1, 2) and (4, 8) on a graph. Find its gradient.

Solution:

Vertical change: 8 - 2 = 6 (go up 6 units)

Horizontal change: 4 - 1 = 3 (go right 3 units)

Gradient = ⁶⁄₃ = 2

Signs:

Going UP = positive vertical change
Going DOWN = negative vertical change
Going RIGHT = positive horizontal change

📝 Finding y-intercept

y-intercept: The point where the line crosses the y-axis. At this point, x = 0.

Example 2

Find the y-intercept of y = 3x - 5

Solution:

Substitute x = 0: y = 3(0) - 5 = -5

y-intercept is (0, -5)

Or: In y = mx + c form, c = -5, so y-intercept is (0, -5)

Example 3

A line has gradient 2 and passes through the point (3, 8). Find the y-intercept.

Solution:

Use y = mx + c: 8 = 2(3) + c

8 = 6 + c

c = 2

y-intercept is (0, 2)

📝 Finding x-intercept

x-intercept: The point where the line crosses the x-axis. At this point, y = 0.

Example 4

Find the x-intercept of y = 2x - 6

Solution:

Set y = 0: 0 = 2x - 6

2x = 6

x = 3

x-intercept is (3, 0)

📝 Interpreting Gradient in Context

Real-world meaning: The gradient represents the rate of change between the two variables.

Example 5

A graph shows the cost (£C) of hiring a car for x days. The equation is C = 40x + 50.

a) What does the gradient represent?

b) What does the y-intercept represent?

Solution:

a) Gradient = 40. This represents £40 per day (daily hire rate)

b) y-intercept = 50. This represents a £50 fixed fee (insurance/admin)

📝 Writing Equation from Graph

Method:

Find the gradient (m) from the graph
Find the y-intercept (c) from the graph
Write the equation: y = mx + c

Example 6

A line on a graph passes through (0, 3) and (2, 9). Write its equation.

Solution:

y-intercept (0, 3), so c = 3

Gradient = ^{9 - 3}⁄_{2 - 0} = ⁶⁄₂ = 3

Equation: y = 3x + 3

🧮 Worked Examples — Exam Style

Straight lines: y = mx + c. m = gradient (steepness, rise/run), c = y-intercept (where it crosses the y-axis).

Find the gradient of the line through (1, 3) and (5, 15).

m = (15 − 3)/(5 − 1) = 12/4 = 3

Rise over run — y-difference over x-difference, consistently ordered.

M1 for the difference quotient, A1 for 3. Swapping one coordinate's order flips the sign — take both differences the same way round.

A line has gradient 2 and passes through (3, 8). Find its equation.

y = 2x + c → substitute (3, 8): 8 = 6 + c → c = 2

y = 2x + 2

M1 for substituting to find c, A1 for y = 2x + 2. Check with the other given point if there is one.

Exam wording to expect: "Find the gradient", "Find the equation of the line", "What does the gradient represent?".

❓ Practice Questions

Q1: Find the gradient of the line passing through (0, 2) and (4, 10).

Q2: Find the y-intercept of y = 5x - 8.

Q3: Find the x-intercept of y = 3x + 9.

Q4: A line has gradient -2 and passes through (1, 5). Find its y-intercept.

Q5: A distance-time graph shows d = 50t + 100. What does the gradient represent?

Q6: Write the equation of a line with gradient 4 and y-intercept (0, -1).

Q7: A line passes through (0, −2) and (6, 10). Write its equation.

Q8: Line y = −x + 4: write the equation of a parallel line through (0, 9).

✅ Answers

2
(0, -8)
(-3, 0)
(0, 7)
Speed = 50 (units per hour)
y = 4x - 1

📋 Model Answers — Full Working

Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.

Q1. m = (10 − 2)/(4 − 0) = 2

Q2. y-intercept = (0, −8) — the constant term

Q3. 0 = 3x + 9 → x = −3, point (−3, 0)

Q4. 5 = −2(1) + c → c = 7 → (0, 7)

Q5. The gradient 50 = speed (50 miles per hour); intercept 100 = starting distance

Q6. y = 4x − 1

Q7. m = (10 − (−2))/(6 − 0) = 2, c = −2 → y = 2x − 2

Q8. Parallel → same gradient −1, through (0, 9) → y = −x + 9

📌 Key Facts to Memorise

y = mx + c: m = gradient, c = y-intercept
Gradient = (y₂ − y₁)/(x₂ − x₁) — rise over run
Parallel lines: SAME gradient; perpendicular: gradients multiply to −1
x-intercept: set y = 0 and solve; y-intercept: set x = 0
Context: gradient = rate per unit; intercept = starting value

🗺️ In the Exam

Straight-line graphs: 3–5 marks per paper, from plotting to equations to interpretation. Real-world gradient reading (cost, speed, filling) is the AO2 layer.

Links to: real-life graphs (A14), coordinate geometry (G11), proportion graphs (R8).

🧭 Board Focus — AQA

Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.

Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.

🎯 Exam Tips

y-intercept: substitute x = 0 into the equation
x-intercept: substitute y = 0 and solve for x
Gradient = change in y ÷ change in x
Positive gradient = uphill; negative = downhill
In context, gradient often means "rate" (price per unit, speed, etc.)

🧠 Problem-Solving Strategies

Problem-Solving

In real contexts, the gradient represents a rate (e.g., £/hour, km/h) and the y-intercept represents a starting value (e.g., fixed fee, initial distance). Always interpret both in context when answering word problems.

Multi-Step Problem

A bath is being filled. After 2 minutes, the water depth is 15 cm. After 5 minutes, it is 30 cm. (a) Find the gradient and interpret it. (b) Find the y-intercept and interpret it. (c) How deep is the bath after 10 minutes?

Solution:

(a) m = ^30-15⁄_5-2 = 5 cm/min (water rises at 5 cm per minute)

(b) y = 5x + c → 15 = 5(2) + c → c = 5 cm (water already 5 cm deep when timing started)

⚠️ Common Errors

Watch Out!

1. Wrong: The y-intercept of y = 5x - 3 is 3 Correct: The y-intercept is -3 (the sign is included: y-intercept at (0, -3))

2. Wrong: Gradient = ^{horizontal change}⁄_{vertical change} Correct: Gradient = ^{vertical change}⁄_{horizontal change} = ^Δy⁄_Δx

3. Wrong: In C = 30t + 50, the gradient means "30 hours per £" Correct: The gradient 30 means £30 per hour (units of y per unit of x)

✍️ 6-Mark Exam Question

Extended Answer

6 marks: Two water tanks are being drained. Tank A: d = 120 - 8t. Tank B: d = 100 - 5t. d = depth in cm, t = time in minutes. (a) Which tank starts fuller? (b) Which drains faster? (c) After how many minutes is Tank A empty? (d) Which tank still has water left after 15 minutes?

(a) Tank A starts at 120 cm, Tank B at 100 cm. Tank A is fuller.

(b) Tank A drains at 8 cm/min (gradient = -8). Tank B drains at 5 cm/min. Tank A drains faster.

(d) After 15 minutes: Tank A = 120 - 8(15) = 0 cm (empty). Tank B = 100 - 5(15) = 25 cm. Tank B still has water.

Mark scheme: (a) 1 mark. (b) 1 mark with reasoning. (c) 2 marks. (d) 2 marks for both tanks checked.

📊 AO3: Reason & Interpret

Reasoning and Interpretation

A monthly phone bill is B = 0.05t + 12, where B = cost (£), t = number of texts.

(a) Explain what the gradient and y-intercept represent.

(b) If the bill is £27, how many texts were sent?

(c) A rival network charges B = 0.08t + 8. After how many texts is it cheaper to stay with the original network?

Answers: (a) Gradient 0.05 = 5p per text. y-intercept 12 = £12 fixed monthly charge. (b) 27 = 0.05t + 12 → 0.05t = 15 → t = 300 texts. (c) 0.05t + 12 < 0.08t + 8 → 4 < 0.03t → t > 133.3. Original network cheaper after 134+ texts.

📝 Exam Questions by Topic

AQA Past Papers - Topic Questions -- AQA
Edexcel Past Papers - Topic Questions -- Edexcel
OCR Past Papers - Topic Questions -- OCR
Eduqas Past Papers - Topic Questions -- Eduqas
CCEA Past Papers - Topic Questions -- CCEA

A10: Gradients & Intercepts

📋 Key Concepts

📝 Finding Gradient from a Graph

📝 Finding y-intercept

📝 Finding x-intercept

📝 Interpreting Gradient in Context

📝 Writing Equation from Graph

🧮 Worked Examples — Exam Style

❓ Practice Questions

✅ Answers

📋 Model Answers — Full Working

📌 Key Facts to Memorise

🗺️ In the Exam

🧭 Board Focus — AQA

🎯 Exam Tips

🧠 Problem-Solving Strategies

Problem-Solving

⚠️ Common Errors

Watch Out!

✍️ 6-Mark Exam Question

Extended Answer

📊 AO3: Reason & Interpret

Reasoning and Interpretation

📝 Exam Questions by Topic

🎬 Video Resources

Ready to ace your GCSE Mathematics exams?

🧠 Flashcards (Spaced Repetition)

📝 Exam Questions by Topic

🎯 Target Tests (Auto-Graded)

🎓 Smart Lesson (Guided)

A10: Gradients & Intercepts

📋 Key Concepts

📝 Finding Gradient from a Graph

📝 Finding y-intercept

📝 Finding x-intercept

📝 Interpreting Gradient in Context

📝 Writing Equation from Graph

🧮 Worked Examples — Exam Style

❓ Practice Questions

✅ Answers

📋 Model Answers — Full Working

📌 Key Facts to Memorise

🗺️ In the Exam

🧭 Board Focus — AQA

🎯 Exam Tips

🧠 Problem-Solving Strategies

Problem-Solving

⚠️ Common Errors

Watch Out!

✍️ 6-Mark Exam Question

Extended Answer

📊 AO3: Reason & Interpret

Reasoning and Interpretation

📝 Exam Questions by Topic

🎬 Video Resources

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Ready to ace your GCSE Mathematics exams?

🧠 Flashcards (Spaced Repetition)

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