IGCSE Maths · Edexcel 4XMA1 · Unit 2 Higher · Graphs

Transformations of Graphs

If you know the shape of y = f(x), you can sketch y = f(x) + 3, y = f(x − 2) or y = 2f(x) without plotting a single point. This page covers the four transformations in the Edexcel specification, how they move points, and the related skills of finding gradients with tangents and solving equations from graphs.

1Translations: y = f(x) + a and y = f(x + a)

Adding a number outside the function moves the graph up or down. Adding it inside the brackets moves it left or right, in the opposite direction to the sign.

Equation	Transformation	Vector
y = f(x) + a	Translation up by a	(0, a)
y = f(x + a)	Translation left by a	(−a, 0)
y = f(x − a)	Translation right by a	(a, 0)

━ y = x² ━ y = x² + 2 (up 2) ━ y = (x − 3)² (right 3)

Key ideaInside the bracket, the graph does the opposite of what you might expect: f(x − 3) moves right 3, because x must be 3 bigger to give the same output.

2Stretches: y = af(x) and y = f(ax)

Equation	Transformation
y = af(x)	Stretch parallel to the y-axis, scale factor a (y-values × a)
y = f(ax)	Stretch parallel to the x-axis, scale factor 1a (x-values ÷ a)

━ y = sin x ━ y = 2 sin x ━ y = sin 2x

y = 2 sin x is twice as tall (maximum 2). y = sin 2x is squashed horizontally: it completes a full cycle by 180° instead of 360°.

3Reflections: y = −f(x) and y = f(−x)

Equation	Transformation
y = −f(x)	Reflection in the x-axis (y-values change sign)
y = f(−x)	Reflection in the y-axis (x-values change sign)

━ y = f(x) ━ y = −f(x) ━ y = f(−x)

4Following a point through a transformation

Exam questions often give one point, such as a turning point, and ask where it goes. Use this table.

Graph	(p, q) moves to	Example: (2, 5) →
y = f(x) + a	(p, q + a)	y = f(x) + 3 → (2, 8)
y = f(x + a)	(p − a, q)	y = f(x + 3) → (−1, 5)
y = af(x)	(p, aq)	y = 3f(x) → (2, 15)
y = f(ax)	(p ÷ a, q)	y = f(2x) → (1, 5)
y = −f(x)	(p, −q)	(2, −5)
y = f(−x)	(−p, q)	(−2, 5)

Worked example

f(x) = x² − 4x. Write g(x) = f(x + 2) as simply as possible, and describe the transformation.

Replace every x with (x + 2): g(x) = (x + 2)² − 4(x + 2).
= x² + 4x + 4 − 4x − 8 = x² − 4.
f(x + 2) is a translation 2 units to the left. Check: f has turning point (2, −4); g = x² − 4 has turning point (0, −4) ✓

g(x) = x² − 4; translation by (−2, 0)

5Transforming trig graphs

The same rules apply to y = sin x and y = cos x, with x in degrees.

Worked example

Describe how the graph of y = cos x is transformed to give y = cos(x − 60) + 2, and find the coordinates of the maximum point of the new graph for 0 ≤ x ≤ 360.

(x − 60) inside: translation 60 to the right. +2 outside: translation 2 up. Together, the translation vector is (60, 2).
y = cos x has a maximum at (0, 1) and at (360, 1).
(0, 1) moves to (60, 3), which lies in the range. (360, 1) would move to (420, 3), which is outside it.

Translation by (60, 2); maximum (60, 3)

━ y = cos x ━ y = cos(x − 60) + 2

6Gradients of curves using tangents

The gradient of a curve changes from point to point. To estimate it at a point, draw a tangent there: a straight line that just touches the curve. Then find the gradient of the tangent using rise ÷ run.

Worked example

Estimate the gradient of y = x² − 2x at the point (3, 3).

Draw the tangent at (3, 3). It passes through about (2, −1) and (4, 7).
Gradient = rise ÷ run = (7 − (−1)) ÷ (4 − 2) = 8 ÷ 2.

Gradient ≈ 4

Exam tipUse two points on the tangent that are far apart. It makes your reading more accurate. Edexcel accepts a range of answers, but the line must look like a tangent.

7Solving equations using graphs

The solutions of f(x) = g(x) are the x-coordinates where the graphs of y = f(x) and y = g(x) cross.

Worked example

The graph of y = x² − 2x − 3 is drawn. Which straight line should you draw to solve x² − 3x − 4 = 0?

Rearrange the equation so one side matches the curve: x² − 3x − 4 = 0 → x² − 2x − 3 = x + 1.
So draw the line y = x + 1 and read off where it crosses the curve.
They cross at x = −1 and x = 4.

Draw y = x + 1; x = −1 or x = 4

Common mistakes

Moving f(x + a) to the right. It moves left by a.
Thinking f(2x) doubles the width. It halves the x-values (scale factor ½).
Applying a vertical change to the x-coordinate of a point, or a horizontal change to the y-coordinate.
Drawing a tangent that cuts through the curve, or using two points very close together.
Reading y-values instead of x-values when solving equations from graphs.

★Key Vocabulary Recap

Translation: A slide, described by a column vector.
Stretch: Scaling parallel to an axis by a scale factor.
Reflection: A mirror image in an axis.
Tangent: A line that touches a curve at one point without crossing it there.
Turning point: A maximum or minimum point of a curve.

Specification reference (Pearson Edexcel 4XMA1, Unit 2 Higher)

Apply to the graph of y = f(x) the transformations y = f(x) + a, y = f(ax), y = f(x + a) and y = af(x) for linear, quadratic, sine and cosine functions.
Interpret and analyse transformations of functions and write them algebraically.
Find the gradients of non-linear graphs by drawing tangents.
Find the intersection points of two graphs, one linear and one non-linear, and recognise that the solutions are the solutions of an equation.

Next stepReady to practise? Try the 12 exam-style questions, then check your working against the worked solutions.

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