Step-by-step worked examples and graded practice questions on graph transformations — translating, reflecting and stretching graphs using function notation.
📚 Higher✅ 15 Practice Questions🔍 Full Worked Examples⚠️ Common Mistakes
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If a graph is written as y = f(x), we can describe transformations of that graph using function notation, without needing to know the actual equation. Throughout this page, we'll apply each transformation to the same graph of y = f(x), which passes through three key points:
A (0, 1)
B (2, 5) — a maximum point
C (4, 1)
Tracking what happens to these three points is the fastest way to sketch any transformed graph.
The graph of y = f(x), with key points A (0, 1), B (2, 5) and C (4, 1).
Vertical translation: y = f(x) + a
Adding a number outside the function shifts the whole graph up (if a is positive) or down (if a is negative). Every y-coordinate increases by a; the x-coordinates don't change.
Worked Example 1 — Translating vertically
Sketch y = f(x) + 3, and state the new coordinates of points A, B and C.
1
Add 3 to every y-coordinate; the x-coordinates stay the same
2
A (0, 1) → (0, 4). B (2, 5) → (2, 8). C (4, 1) → (4, 4)
Answer(0, 4), (2, 8), (4, 4) — the graph moves up 3 units
y = f(x) + 3 (solid) shifts every point up 3 units from y = f(x) (dashed).
Horizontal translation: y = f(x + a)
Adding a number inside the brackets shifts the graph horizontally — but in the opposite direction to what you might expect. y = f(x − a) shifts the graph right by a; y = f(x + a) shifts the graph left by a.
Worked Example 2 — Translating horizontally
Sketch y = f(x − 2), and state the new coordinates of points A, B and C.
1
y = f(x − 2) shifts the graph 2 units to the right: add 2 to every x-coordinate
2
A (0, 1) → (2, 1). B (2, 5) → (4, 5). C (4, 1) → (6, 1)
Answer(2, 1), (4, 5), (6, 1) — the graph moves right 2 units
y = f(x − 2) (solid) shifts every point 2 units to the right of y = f(x) (dashed).
Reflections: y = −f(x) and y = f(−x)
y = −f(x) reflects the graph in the x-axis — every y-coordinate changes sign. y = f(−x) reflects the graph in the y-axis — every x-coordinate changes sign.
Worked Example 3 — Reflecting in the x-axis
Sketch y = −f(x), and state the new coordinates of points A, B and C.
1
Change the sign of every y-coordinate; the x-coordinates stay the same
2
A (0, 1) → (0, −1). B (2, 5) → (2, −5). C (4, 1) → (4, −1)
Answer(0, −1), (2, −5), (4, −1) — the maximum point B becomes a minimum
y = −f(x) (solid) reflects y = f(x) (dashed) in the x-axis — the maximum becomes a minimum.
Worked Example 4 — Reflecting in the y-axis
Sketch y = f(−x), and state the new coordinates of points A, B and C.
1
Change the sign of every x-coordinate; the y-coordinates stay the same
2
A (0, 1) → (0, 1) (unchanged, since 0 has no sign to flip). B (2, 5) → (−2, 5). C (4, 1) → (−4, 1)
Answer(0, 1), (−2, 5), (−4, 1)
y = f(−x) (solid) reflects y = f(x) (dashed) in the y-axis.
Stretches: y = af(x) and y = f(ax)
y = af(x) stretches the graph vertically by scale factor a, measured from the x-axis — multiply every y-coordinate by a. y = f(ax) stretches the graph horizontally by scale factor 1/a, measured from the y-axis — divide every x-coordinate by a.
Worked Example 5 — Vertical and horizontal stretches
Using point B (2, 5), find the corresponding point on (a) y = 3f(x), and (b) y = f(2x).
1
(a) y = 3f(x): multiply the y-coordinate by 3. B (2, 5) → (2, 15)
2
(b) y = f(2x): divide the x-coordinate by 2. B (2, 5) → (1, 5)
Answer(a) (2, 15) (b) (1, 5)
y = 3f(x) (teal) stretches vertically; y = f(2x) (amber) squashes horizontally — both compared to y = f(x) (dashed).
Practice questions
Unless stated otherwise, questions refer to a graph y = f(x) with the given point(s) on it. Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1The graph y = f(x) has a maximum at (3, 4). State the coordinates of the maximum on y = f(x) + 5.Foundation
Q2The point (2, 7) lies on y = f(x). State the coordinates of the corresponding point on y = f(x) − 3.Foundation
Q3The point (5, 2) lies on y = f(x). State the coordinates of the corresponding point on y = f(x − 4).Foundation
Q4The point (6, 1) lies on y = f(x). State the coordinates of the corresponding point on y = f(x + 2).Foundation
Q5The graph y = f(x) has a minimum at (1, −3). State the coordinates of the minimum on y = −f(x).Foundation
Higher (Grade 5–7)
Q6The point (4, 6) lies on y = f(x). State the coordinates of the corresponding point on y = f(−x).Higher
Q7The point (3, 8) lies on y = f(x). State the coordinates of the corresponding point on y = 2f(x).Higher
Q8The point (6, 9) lies on y = f(x). State the coordinates of the corresponding point on y = f(3x).Higher
Q9The point (2, 5) lies on y = f(x). State the coordinates of the corresponding point on y = f(x/2).Higher
Q10The graph y = f(x) has a maximum at (2, 5). Describe fully the single transformation that maps y = f(x) to y = f(x) + 4, and state the new maximum point.Higher
Higher — Hard (Grade 8–9)
Q11Using the points A (0, 1), B (2, 5) and C (4, 1) from y = f(x), find the coordinates of A, B and C on y = f(x − 1) + 2.Grade 8–9
Q12Using the same points, find the coordinates of A, B and C on y = −2f(x).Grade 8–9
Q13A curve y = f(x) has a root (x-intercept) at x = 6. Find the x-intercept of y = f(x + 4).Grade 8–9
Q14A curve y = f(x) has a root at x = −3. Find the root of y = f(2x).Grade 8–9
Q15The graph of y = f(x) has a single maximum point at (3, 7). The graph of y = f(x + a) − b has its maximum at (1, 2). Find the values of a and b.Grade 8–9
Answers
Foundation (Q1–Q5)
Q1(3, 9)(4 + 5)
Q2(2, 4)(7 − 3)
Q3(9, 2)(f(x−4) shifts right 4: 5 + 4 = 9)
Q4(4, 1)(f(x+2) shifts left 2: 6 − 2 = 4)
Q5(1, 3)(sign of y flips: −(−3) = 3)
Higher (Q6–Q10)
Q6(−4, 6)(sign of x flips)
Q7(3, 16)(8 × 2)
Q8(2, 9)(x-coordinate ÷ 3: 6 ÷ 3 = 2)
Q9(4, 5)(x-coordinate × 2)
Q10Translation by vector (0, 4) — a vertical shift up 4 units. New maximum: (2, 9)
Higher — Hard (Q11–Q15)
Q11A → (1, 3), B → (3, 7), C → (5, 3)(shift right 1, then up 2: x+1, y+2)
Q12A → (0, −2), B → (2, −10), C → (4, −2)(y-values × −2)
Q13x = 2(f(x+4) shifts left 4: 6 − 4 = 2)
Q14x = −1.5(f(2x) halves x-coordinates: −3 ÷ 2)
Q15a = 2, b = 5(new x = old x − a: 1 = 3 − a → a = 2. new y = old y − b: 2 = 7 − b → b = 5)
Common mistakes
These are the errors Alamin sees most frequently with graph transformations at GCSE. Recognising them now will save you marks in the exam.
Common Mistake 1
Getting the horizontal translation direction backwards
This is the single most common error in the whole topic. y = f(x − a) shifts RIGHT, and y = f(x + a) shifts LEFT — the opposite of what the sign suggests. Vertical translations (y = f(x) + a) work the way you'd expect; horizontal ones don't.
Common Mistake 2
Mixing up y = af(x) and y = f(ax)
y = af(x) multiplies the y-coordinates (vertical stretch). y = f(ax) divides the x-coordinates (horizontal stretch/squash). Applying the scale factor to the wrong axis is a very common slip.
Common Mistake 3
Forgetting that y = f(ax) divides (not multiplies) the x-coordinate
For y = f(2x), the graph is squashed horizontally — each x-coordinate is divided by 2, not multiplied. It's easy to assume "2x" means "multiply by 2", but the transformation works in reverse for the x-axis.
Common Mistake 4
Reflecting in the wrong axis
y = −f(x) reflects in the x-axis (y-coordinates flip sign). y = f(−x) reflects in the y-axis (x-coordinates flip sign). Students often reflect in the axis suggested by which variable has the minus sign, which is exactly backwards.
Common Mistake 5
Applying combined transformations in the wrong order
For y = f(x + a) − b, apply the horizontal shift first (using the rule inside the brackets), then the vertical shift (the term outside). Mixing up the order can give the wrong final coordinates, especially when a stretch is also involved.
Common Mistake 6
Not checking the transformed point against the original shape
After transforming, sense-check: a maximum should still be the highest point (unless reflected in the x-axis), and the overall shape of the curve shouldn't change under a translation or reflection — only stretches change the shape.
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