11-13-2025, 01:40 PM
Graphs & Functions — Quick Guide (GCSE & A-Level)
A simple guide to understanding graphs, functions, gradients, intercepts, and common graph shapes.
Perfect for GCSE maths, A-Level foundations, and interpreting scientific data.
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1. What Is a Function?
A function links each input (x) to exactly one output (y).
Written as:
• y = f(x)
• f(3) means “the value of the function when x = 3”
Examples:
• y = 2x + 1
• y = x²
• y = 1/x
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2. Axes & Coordinates
Points are written as (x, y):
• x = horizontal
• y = vertical
Examples:
• (3, 2) → 3 right, 2 up
• (-1, -4) → left & down
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3. Gradient (Slope)
The gradient of a straight line is:
gradient = change in y ÷ change in x
(between any two points)
Symbol: m
• Positive gradient → line slopes upward
• Negative gradient → line slopes downward
• Gradient = 0 → flat line
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4. Straight Line Graphs (y = mx + c)
m = gradient
c = y-intercept (where the line crosses the y-axis)
Examples:
• y = 3x + 2 → gradient 3, y-intercept 2
• y = -2x + 5 → gradient -2, intercept 5
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5. Quadratic Graphs (y = ax² + bx + c)
Shape: a U-shaped curve called a parabola.
• If a > 0 → opens upward (smile)
• If a < 0 → opens downward (frown)
Features:
• turning point (vertex)
• line of symmetry
• roots → where the graph crosses x-axis
Example: y = x² is the simplest quadratic.
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6. Cubic Graphs (y = x³)
S-shaped curve.
• Falls → then rises (if coefficient positive)
• Symmetrical around the origin
Useful for modelling more complex patterns.
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7. Reciprocal Graphs (y = 1/x)
Two separate curves in opposite corners.
Key features:
• never touches the axes
• “hyperbola” shape
• undefined at x = 0
Used in physics for inverse relationships.
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8. Exponential Graphs (y = aˣ)
• rapid growth or decay
• always positive
• passes through (0,1)
Examples:
• y = 2ˣ
• y = eˣ
Very common in science, population models, and finance.
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9. Interpreting Graphs in Science
Graphs show relationships:
• The steeper the line → the faster the change
• A flat graph → no change
• A curved graph → accelerating or slowing
• A straight graph → constant rate
Examples:
Velocity-time graph:
• gradient = acceleration
• area under graph = distance
Distance-time graph:
• gradient = speed
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10. Domain & Range (A-Level)
• Domain → allowed x-values
• Range → possible y-values
Example:
y = √x
• domain: x ≥ 0
• range: y ≥ 0
-----------------------------------------------------------------------
11. Transformations of Graphs
Quick rules:
• y = f(x) + k → move up
• y = f(x - k) → move right
• y = f(x + k) → move left
• y = -f(x) → reflection in x-axis
• y = f(-x) → reflection in y-axis
• y = k·f(x) → vertical stretch
-----------------------------------------------------------------------
12. Common Mistakes
❌ Confusing gradient with y-intercept
❌ Mixing up distance-time vs velocity-time graphs
❌ Reading graphs “by eye” instead of using coordinates
❌ Forgetting negative gradients
❌ Thinking curves have constant rates
✔ Always:
• read coordinates carefully
• calculate gradients properly
• understand the context of scientific graphs
-----------------------------------------------------------------------
Summary
Key ideas:
• y = mx + c → straight lines
• gradients → rate of change
• quadratics → curved U shape
• reciprocals → two branches
• exponentials → rapid growth/decay
• transformations shift or flip graphs
Master these and most graph questions become easy.
A simple guide to understanding graphs, functions, gradients, intercepts, and common graph shapes.
Perfect for GCSE maths, A-Level foundations, and interpreting scientific data.
-----------------------------------------------------------------------
1. What Is a Function?
A function links each input (x) to exactly one output (y).
Written as:
• y = f(x)
• f(3) means “the value of the function when x = 3”
Examples:
• y = 2x + 1
• y = x²
• y = 1/x
-----------------------------------------------------------------------
2. Axes & Coordinates
Points are written as (x, y):
• x = horizontal
• y = vertical
Examples:
• (3, 2) → 3 right, 2 up
• (-1, -4) → left & down
-----------------------------------------------------------------------
3. Gradient (Slope)
The gradient of a straight line is:
gradient = change in y ÷ change in x
(between any two points)
Symbol: m
• Positive gradient → line slopes upward
• Negative gradient → line slopes downward
• Gradient = 0 → flat line
-----------------------------------------------------------------------
4. Straight Line Graphs (y = mx + c)
m = gradient
c = y-intercept (where the line crosses the y-axis)
Examples:
• y = 3x + 2 → gradient 3, y-intercept 2
• y = -2x + 5 → gradient -2, intercept 5
-----------------------------------------------------------------------
5. Quadratic Graphs (y = ax² + bx + c)
Shape: a U-shaped curve called a parabola.
• If a > 0 → opens upward (smile)
• If a < 0 → opens downward (frown)
Features:
• turning point (vertex)
• line of symmetry
• roots → where the graph crosses x-axis
Example: y = x² is the simplest quadratic.
-----------------------------------------------------------------------
6. Cubic Graphs (y = x³)
S-shaped curve.
• Falls → then rises (if coefficient positive)
• Symmetrical around the origin
Useful for modelling more complex patterns.
-----------------------------------------------------------------------
7. Reciprocal Graphs (y = 1/x)
Two separate curves in opposite corners.
Key features:
• never touches the axes
• “hyperbola” shape
• undefined at x = 0
Used in physics for inverse relationships.
-----------------------------------------------------------------------
8. Exponential Graphs (y = aˣ)
• rapid growth or decay
• always positive
• passes through (0,1)
Examples:
• y = 2ˣ
• y = eˣ
Very common in science, population models, and finance.
-----------------------------------------------------------------------
9. Interpreting Graphs in Science
Graphs show relationships:
• The steeper the line → the faster the change
• A flat graph → no change
• A curved graph → accelerating or slowing
• A straight graph → constant rate
Examples:
Velocity-time graph:
• gradient = acceleration
• area under graph = distance
Distance-time graph:
• gradient = speed
-----------------------------------------------------------------------
10. Domain & Range (A-Level)
• Domain → allowed x-values
• Range → possible y-values
Example:
y = √x
• domain: x ≥ 0
• range: y ≥ 0
-----------------------------------------------------------------------
11. Transformations of Graphs
Quick rules:
• y = f(x) + k → move up
• y = f(x - k) → move right
• y = f(x + k) → move left
• y = -f(x) → reflection in x-axis
• y = f(-x) → reflection in y-axis
• y = k·f(x) → vertical stretch
-----------------------------------------------------------------------
12. Common Mistakes
❌ Confusing gradient with y-intercept
❌ Mixing up distance-time vs velocity-time graphs
❌ Reading graphs “by eye” instead of using coordinates
❌ Forgetting negative gradients
❌ Thinking curves have constant rates
✔ Always:
• read coordinates carefully
• calculate gradients properly
• understand the context of scientific graphs
-----------------------------------------------------------------------
Summary
Key ideas:
• y = mx + c → straight lines
• gradients → rate of change
• quadratics → curved U shape
• reciprocals → two branches
• exponentials → rapid growth/decay
• transformations shift or flip graphs
Master these and most graph questions become easy.