Small changes to an equation move, flip or stretch its graph in predictable ways. Learn the four basic transformations, describe them with the exact words examiners want, and combine them in the right order.
Changes OUTSIDE the function act on \(y\) as you would expect: \(f(x) + a\) moves up \(a\), \(af(x)\) stretches by factor \(a\) parallel to the \(y\)-axis. Changes INSIDE act on \(x\) the opposite way: \(f(x + a)\) moves LEFT \(a\), \(f(ax)\) stretches by factor \(\dfrac{1}{a}\) parallel to the \(x\)-axis.
Fig. 1: y = (x - 3)² + 1 is y = x² translated 3 units right and 1 unit up; every point, such as (0, 0) and (-2, 4), moves by the same vector
Section 01
Translations.
A translation slides the whole graph without changing its shape. Describe it as 'a translation by the vector \(\begin{pmatrix}p\\q\end{pmatrix}\)', where \(p\) is the move in the \(x\)-direction and \(q\) the move in the \(y\)-direction.
Equation
Transformation
Point \((x, y)\) goes to
\(y = f(x) + a\)
translation \(\begin{pmatrix}0\\a\end{pmatrix}\)
\((x,\ y + a)\)
\(y = f(x + a)\)
translation \(\begin{pmatrix}-a\\0\end{pmatrix}\)
\((x - a,\ y)\)
\(y = f(x - p) + q\)
translation \(\begin{pmatrix}p\\q\end{pmatrix}\)
\((x + p,\ y + q)\)
Worked example Describe a translation
Describe the transformation that maps \(y = x^{2}\) onto \(y = (x - 3)^{2} + 1\) (Fig. 1).
\(x\) is replaced by \(x - 3\): the graph moves 3 units in the positive \(x\)-direction.
\(+1\) is added outside: the graph moves 1 unit up.
Check with the vertex: \((0, 0)\) moves to \((3, 1)\), the vertex of the new curve.
Answer: A translation by \(\begin{pmatrix}3\\1\end{pmatrix}\)
Worked example Find the new equation
The curve \(y = x^{2} + 2x\) is translated by \(\begin{pmatrix}2\\-3\end{pmatrix}\). Find the equation of the new curve in the form \(y = x^{2} + bx + c\).
Replace \(x\) by \(x - 2\) and subtract 3: \(y = (x - 2)^{2} + 2(x - 2) - 3\).
Expand: \(x^{2} - 4x + 4 + 2x - 4 - 3\).
Answer: \(y = x^{2} - 2x - 3\)
Examiner trap
\(y = f(x + 4)\) moves the graph 4 units LEFT, not right. A point \((2, 5)\) on \(y = f(x)\) goes to \((-2, 5)\): you need \(x + 4 = 2\) to get the same output.
Section 02
Reflections and stretches.
Reflections flip the graph in an axis; stretches pull it away from an axis. A full description of a stretch states the direction (parallel to which axis) AND the scale factor.
Equation
Transformation
\((x, y)\) goes to
\(y = -f(x)\)
reflection in the \(x\)-axis
\((x,\ -y)\)
\(y = f(-x)\)
reflection in the \(y\)-axis
\((-x,\ y)\)
\(y = af(x)\)
stretch parallel to the \(y\)-axis, factor \(a\)
\((x,\ ay)\)
\(y = f(ax)\)
stretch parallel to the \(x\)-axis, factor \(\frac{1}{a}\)
\(\left(\frac{x}{a},\ y\right)\)
Worked example Following a point
The point \((4, -6)\) lies on \(y = f(x)\). Find the image of this point on (a) \(y = 3f(x)\) (b) \(y = f(2x)\) (c) \(y = f(-x)\).
(a) Stretch parallel to the \(y\)-axis, factor 3: multiply \(y\) by 3, giving \((4, -18)\).
Describe a single transformation that maps \(y = x^{3}\) onto \(y = 8x^{3}\).
\(8x^{3} = 8f(x)\) with \(f(x) = x^{3}\): a stretch parallel to the \(y\)-axis, factor 8.
Also \(8x^{3} = (2x)^{3} = f(2x)\): a stretch parallel to the \(x\)-axis, factor \(\frac{1}{2}\).
Either description is correct, because the two equations are identical.
Answer: A stretch parallel to the \(y\)-axis with scale factor 8 (or parallel to the \(x\)-axis with factor \(\frac{1}{2}\))
Examiner trap
\(y = f(2x)\) is a stretch by factor \(\frac{1}{2}\), not 2: the graph is squeezed towards the \(y\)-axis. And always say 'parallel to the \(x\)-axis' or 'parallel to the \(y\)-axis'; 'a stretch of factor 2' alone scores nothing.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Describe the transformation from \(y = f(x)\) to \(y = f(x) - 4\).
Quick check+10 XP
Describe the transformation from \(y = f(x)\) to \(y = f(x + 2)\).
Quick check+10 XP
Describe the transformation from \(y = f(x)\) to \(y = f(3x)\).
Quick check+10 XP
\((2, 7)\) lies on \(y = f(x)\). Which point lies on \(y = -f(x)\)?
Section 03
Combining transformations.
Two transformations in DIFFERENT directions (one on \(x\), one on \(y\)) can be done in either order. Two in the SAME direction usually cannot: build the equation step by step and check.
Worked example Order on the outside
Describe a sequence of two transformations that maps \(y = f(x)\) onto \(y = 2f(x) + 3\).
Stretch first: \(y = 2f(x)\).
Then translate up 3: \(y = 2f(x) + 3\). This order works.
The other order fails: translating first gives \(f(x) + 3\), and stretching that gives \(2f(x) + 6\).
Answer: A stretch parallel to the \(y\)-axis, factor 2, followed by a translation by \(\begin{pmatrix}0\\3\end{pmatrix}\)
Worked example Order on the inside
Describe a sequence of two transformations that maps \(y = f(x)\) onto \(y = f(2x - 6)\).
Option 1, translate first: \(y = f(x - 6)\), then replace \(x\) by \(2x\): \(f(2x - 6)\). It works.
Option 2, stretch first: \(y = f(2x)\), then replace \(x\) by \(x - 3\): \(f\big(2(x - 3)\big) = f(2x - 6)\). It also works, with a different vector.
Translating by 3 then stretching would give \(f(2x - 3)\), which is wrong.
Answer: A translation by \(\begin{pmatrix}6\\0\end{pmatrix}\) followed by a stretch parallel to the \(x\)-axis, factor \(\frac{1}{2}\) (or the stretch first, then a translation by \(\begin{pmatrix}3\\0\end{pmatrix}\))
Worked example Build an equation
The curve \(y = x^{2} - 4x\) is reflected in the \(x\)-axis and then translated by \(\begin{pmatrix}0\\5\end{pmatrix}\). Find the equation of the resulting curve.
Reflect in the \(x\)-axis: \(y = -(x^{2} - 4x) = -x^{2} + 4x\).
Translate up 5: add 5.
Answer: \(y = -x^{2} + 4x + 5\)
Your turn
The point \((6, 4)\) lies on \(y = f(x)\). Find its image on \(y = 2f\left(\dfrac{x}{3}\right)\).
\(f\left(\frac{x}{3}\right)\) is a stretch parallel to the \(x\)-axis, factor 3, so \(x\) becomes 18; the factor 2 outside doubles \(y\). The image is \((18, 8)\).
Examiner trap
Inside the bracket, translations and stretches interact: \(f(2x - 6)\) is NOT 'translate 6 right, stretch factor \(\frac{1}{2}\)' unless you do them in that order. Test any sequence by building the equation.
Stage 2 · Exam
Exam-style questions
Unlocks once the four checks above are done. Worth more XP, written to AS & A Level Paper 1 to 6 standard.
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Exam style+20 XP
The curve \(y = x^{2} + 2x\) is translated by \(\begin{pmatrix}2\\-3\end{pmatrix}\). Find the equation of the new curve.
Exam style+20 XP
The point \((6, 4)\) lies on \(y = f(x)\). Find the corresponding point on \(y = 2f\left(\dfrac{x}{3}\right)\).
Exam style+20 XP
Which sequence of transformations maps \(y = f(x)\) onto \(y = 3f(x) - 2\)?
Exam style+20 XP
Which sequence of transformations maps \(y = f(x)\) onto \(y = f(2x + 4)\)?
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Graphs moved, flipped and stretched.
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