Transformations of Graphs of Functions
Transformations of graphs of functions change the position or the shape of a curve without changing the rule it came from, and IGCSE exam questions test whether the two families are kept apart. Changes OUTSIDE the function affect the y values in a predictable direction: y = f(x) + a translates the curve a units up, and y = af(x) stretches it by scale factor a parallel to the y axis, while y = -f(x) reflects it in the x axis. Changes INSIDE the bracket affect the x values and behave in the opposite way to how they read: y = f(x + a) translates the curve a units to the LEFT, y = f(ax) stretches it by scale factor 1/a parallel to the x axis, and y = f(-x) reflects it in the y axis.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Decide first whether the change is inside or outside the bracket, because that single question determines which axis is affected and whether the effect is intuitive or reversed.
- Outside the bracket affects y and does what it says: adding a moves the graph up by a, multiplying by a stretches it vertically by a, and a minus sign reflects it in the x axis.
- Inside the bracket affects x and does the opposite of what it says: f(x + a) moves LEFT by a, f(x - a) moves RIGHT by a, f(ax) squashes horizontally by a factor of 1/a, and f(-x) reflects in the y axis.
- Apply transformations to key points rather than trying to picture the whole curve: take the vertex, the intercepts and any turning point, transform each one, and sketch through the images.
- For a translation, quote the column vector form when the question asks for a description, since that is what earns the mark.
- For a combined transformation, apply the inside change and the outside change separately and say which order you used; for points, the horizontal change acts on the x coordinate and the vertical change on the y coordinate, so they do not interfere.
Worked example
The curve y = f(x) has a minimum at (1, -4). Find the coordinates of the minimum of (a) y = f(x) + 3, (b) y = f(x - 2), (c) y = 2f(x), and (d) y = f(-x).
- For y = f(x) + 3 the change is outside, so it moves the curve 3 up: the x coordinate is unchanged and the y coordinate increases by 3, giving (1, -1).
- For y = f(x - 2) the change is inside, so it moves the curve right by 2, opposite to the minus sign's appearance: the minimum moves to (3, -4).
- For y = 2f(x) the change is outside and multiplies every y value by 2: the minimum moves to (1, -8).
- For y = f(-x) the change is inside and reflects the curve in the y axis: the x coordinate changes sign, giving (-1, -4).
- Check the pattern: in (a) and (c) only the y coordinate changed, and both are outside changes; in (b) and (d) only the x coordinate changed, and both are inside changes.
- Final answers: (a) (1, -1), (b) (3, -4), (c) (1, -8), (d) (-1, -4).
Practice questions
Try each question, then tap to reveal the answer.
Q1Describe the transformation from y = f(x) to y = f(x) - 5.Show answer
Answer: A translation 5 units down, that is by the column vector 0 over -5.
Q2Describe the transformation from y = f(x) to y = f(x + 4).Show answer
Answer: A translation 4 units to the left, by the column vector -4 over 0.
Q3Why does f(x + a) move the graph left rather than right?Show answer
Answer: The curve reaches each output at a smaller x than before, because a is added before the function acts, so every point occurs a units earlier along the x axis.
Q4The point (2, 5) lies on y = f(x). Find its image on y = 3f(x).Show answer
Answer: (2, 15), since only the y coordinate is multiplied.
Q5The point (6, -1) lies on y = f(x). Find its image on y = f(2x).Show answer
Answer: (3, -1), since the horizontal stretch has scale factor one half.
Q6Describe the transformation from y = f(x) to y = -f(x).Show answer
Answer: A reflection in the x axis.
Q7The curve y = x squared is translated to give a vertex at (4, 1). Write its equation.Show answer
Answer: y = (x - 4) squared + 1.
Exam-style questions
Written in the style of a IGCSE Maths exam paper, with a full mark scheme.
The curve y = f(x) passes through (0, 3) and has a maximum at (2, 7). Sketch is not required. Find, for the curve y = f(x + 1) - 4, (a) the coordinates of the maximum, and (b) the y coordinate where x = -1.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 5 available
Explain, using the idea of inside and outside changes, why y = f(2x) squashes the graph horizontally while y = 2f(x) stretches it vertically.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 4 available
Free printable worksheet
Want more practice on paper? Download the transformations of graphs of functions worksheet pack - 6 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
Next topics
Not quite what you needed?
Tell us what is missing on transformations of graphs of functions, or which topic to write up next. Every request is read, and we reply to every one.
Build a full practice pack.
This topic is one of hundreds in the library - pick the ones a student needs and generate a printable PDF in minutes.