Graphs & transformations at A-Level
You already know how to apply a single transformation to a graph, and the basic shapes of cubic and reciprocal graphs from GCSE. A-Level pushes further: sketching cubics directly from factorised form using root behaviour, a more formal treatment of reciprocal graphs and their asymptotes, finding where two graphs intersect, and — the skill GCSE never asks for — combining two transformations in a single question, in the correct order.
Four skills make up this topic:
- Cubic graphs from factorised form — using each root's multiplicity to decide whether the curve crosses or touches
- Reciprocal graphs — \(y = \dfrac{1}{x}\) and \(y = \dfrac{1}{x^2}\), with their asymptotes stated formally
- Points of intersection — finding exactly where two graphs meet
- Combining transformations — applying two transformations to the same graph, in the right order
Sketching cubics from factorised form
Once a cubic is factorised, each root tells you how the curve behaves there: a single root means the curve crosses straight through the x-axis; a repeated (squared) root means the curve touches the x-axis and turns back, without crossing.
Worked Example 1
Sketch \(y = (x - 2)(x + 1)(x - 3)\), showing where the curve crosses the x-axis.
1
Three distinct roots: \(x = 2\), \(x = -1\), \(x = 3\) — the curve crosses the x-axis at each one
2
The coefficient of \(x^3\) is positive, so the curve rises from bottom-left to top-right overall
3
y-intercept: substitute \(x = 0\): \(y = (-2)(1)(-3) = 6\)
AnswerCrosses at \(x = -1, 2, 3\); y-intercept \((0, 6)\)
Worked Example 2
Sketch \(y = (x + 1)^2(x - 2)\), describing the curve's behaviour at each root.
1
\(x = -1\) is a repeated (squared) root, so the curve touches the x-axis there and turns back
2
\(x = 2\) is a single root, so the curve crosses straight through
3
y-intercept: substitute \(x = 0\): \(y = (1)^2(-2) = -2\)
AnswerTouches at \((-1, 0)\); crosses at \((2, 0)\); y-intercept \((0, -2)\)
The curve \(y = (x + 1)^2(x - 2)\) touches the x-axis at its repeated root \((-1, 0)\) and crosses at its single root \((2, 0)\) — the shapes found in Worked Example 2.
Reciprocal graphs
\(y = \dfrac{1}{x}\) and \(y = \dfrac{1}{x^2}\) never touch the axes — as \(x\) approaches 0, \(y\) grows without bound, and as \(x\) grows without bound, \(y\) approaches 0. Both lines the curve approaches but never reaches are called asymptotes.
Worked Example 3
State the equations of the asymptotes of \(y = \dfrac{1}{x}\), and describe the curve's shape.
1
As \(x \to 0\), \(y \to \pm\infty\), so \(x = 0\) is a vertical asymptote
2
As \(x \to \pm\infty\), \(y \to 0\), so \(y = 0\) is a horizontal asymptote
3
For \(x > 0\), \(y\) is positive (curve in the top-right region); for \(x < 0\), \(y\) is negative (curve in the bottom-left region)
AnswerAsymptotes \(x = 0\) and \(y = 0\); two branches, one in each of those diagonal regions
\(y = \dfrac{1}{x}\) approaches but never reaches either axis — \(x = 0\) and \(y = 0\) are both asymptotes, found in Worked Example 3.
Points of intersection
To find where two graphs meet, set the two expressions for \(y\) equal to each other and solve — exactly the same technique used for linear-quadratic simultaneous equations.
Worked Example 4
Find the points of intersection of \(y = x^2\) and \(y = x + 2\).
1
Set equal: \(x^2 = x + 2\)
2
Rearrange: \(x^2 - x - 2 = 0\)
3
Factorise: \((x - 2)(x + 1) = 0\), so \(x = 2\) or \(x = -1\)
4
Substitute into \(y = x + 2\): \(x = 2\) gives \(y = 4\); \(x = -1\) gives \(y = 1\)
Answer\((2, 4)\) and \((-1, 1)\)
Combining transformations
Read a combined transformation from the inside out: apply whatever is happening to \(x\) first (inside the brackets), then apply whatever is happening to the whole function second (outside the brackets). Track a few known points through both steps in order.
Worked Example 5
The graph of \(y = f(x)\) passes through \(A(1, 2)\), \(B(3, 6)\) and \(C(5, 2)\). Find the coordinates of the corresponding points on \(y = f(x - 2) + 3\).
1
\(x - 2\) translates the graph 2 units right: add 2 to every x-coordinate
2
\(+ 3\) outside translates the graph 3 units up: add 3 to every y-coordinate
3
\(A(1, 2) \to (1+2, 2+3)\); \(B(3, 6) \to (3+2, 6+3)\); \(C(5, 2) \to (5+2, 2+3)\)
Answer\(A'(3, 5)\), \(B'(5, 9)\), \(C'(7, 5)\)
Worked Example 6
The graph of \(y = f(x)\) passes through \(A(1, 2)\), \(B(3, 6)\) and \(C(5, 2)\). Find the coordinates of the corresponding points on \(y = 2f(x + 1)\).
1
Work from the inside out. \(x + 1\) translates the graph 1 unit left: subtract 1 from every x-coordinate
2
The 2 outside the function stretches vertically by factor 2: multiply every y-coordinate by 2
3
\(A(1, 2) \to (1-1, 2 \times 2)\); \(B(3, 6) \to (3-1, 6 \times 2)\); \(C(5, 2) \to (5-1, 2 \times 2)\)
Answer\(A'(0, 4)\), \(B'(2, 12)\), \(C'(4, 4)\)
Exam tips
💡 Tip 1
Squared factor = touches; single factor = crosses
Check the power on each factor before sketching. \((x - a)\) crosses at \(a\); \((x - a)^2\) touches at \(a\); \((x - a)^3\) crosses at \(a\) but flattens out as it does.
💡 Tip 2
State asymptotes as equations, not descriptions
Write "\(x = 0\)" and "\(y = 0\)", not "the y-axis" and "the x-axis" — exam mark schemes look for the equation form specifically.
💡 Tip 3
Always check your intersection points in both original equations
Substituting your \((x, y)\) pair back into both starting equations catches sign and arithmetic errors before you lose marks for a wrong final answer.
💡 Tip 4
Work from the inside out for combined transformations
Whatever is done to \(x\) happens first; whatever is done to the whole function happens second. For \(y = 3f(x - 2)\), translate first, then stretch — never the other way round.
💡 Tip 5
Track named points, not the whole curve
When a question gives you specific points on \(y = f(x)\), transform those points individually rather than trying to redraw the whole curve from scratch — it's faster and far less error-prone.
Common mistakes
Common Mistake 1
Treating every root the same way
For \(y = (x + 1)^2(x - 2)\), sketching the curve crossing straight through \(x = -1\) ignores that the factor is squared. A repeated root means the curve touches and turns back — it never crosses there.
Common Mistake 2
Drawing a reciprocal graph touching the axes
\(y = \dfrac{1}{x}\) gets arbitrarily close to both axes but never actually touches or crosses either one. Drawing the curve meeting an axis is a common and costly sketching error.
Common Mistake 3
Only finding x, and forgetting to find the matching y
"Find the points of intersection" needs full coordinate pairs. Finding \(x = 2\) and \(x = -1\) but not substituting back in for the y-values is an incomplete answer.
Common Mistake 4
Applying combined transformations in the wrong order
For \(y = 2f(x + 1)\), stretching the original points by 2 before translating gives the wrong answer. Always deal with what's inside the brackets (the x-transformation) first, then what's outside.
Common Mistake 5
Forgetting the sign flip for a horizontal translation
\(y = f(x + 1)\) shifts the graph left by 1, not right — the sign inside the bracket is the opposite of the direction of the shift. Only the vertical shift (outside the bracket) matches its sign directly.
Practice questions
Work through each question before checking the worked solution.
Core Skills
Q1Sketch \(y = (x - 1)(x + 2)(x - 4)\), stating where the curve crosses the x-axis.Core Skills
Q2Sketch \(y = x^2(x + 3)\), describing the behaviour of the curve at each root.Core Skills
Q3Write down the equations of the asymptotes of \(y = \dfrac{1}{x}\).Core Skills
Q4Find the points of intersection of \(y = x^2\) and \(y = 3x - 2\).Core Skills
Q5The graph of \(y = f(x)\) passes through \((2, 3)\). State the coordinates of the corresponding point on \(y = f(x) + 4\).Core Skills
Exam-Style
Q6Sketch \(y = (x + 2)(x - 1)^2\), stating where the curve crosses and where it touches the x-axis.Exam-Style
Q7State the equations of the asymptotes of \(y = \dfrac{1}{x^2}\), and explain why the curve has no y-intercept.Exam-Style
Q8Find the points of intersection of \(y = x^2 - 4\) and \(y = 2x - 1\).Exam-Style
Q9The graph of \(y = f(x)\) passes through \((3, -2)\). State the coordinates of the corresponding point on \(y = f(x - 5)\).Exam-Style
Q10The graph of \(y = f(x)\) passes through \((4, 6)\). State the coordinates of the corresponding point on \(y = 3f(x)\).Exam-Style
A* Challenge
Q11Sketch \(y = (x - 2)^2(x + 3)\), and find the coordinates of the y-intercept.A* Challenge
Q12Find the points of intersection of \(y = x^2 - 2x - 3\) and \(y = x - 3\).A* Challenge
Q13The graph of \(y = f(x)\) passes through \((2, 5)\). State the coordinates of the corresponding point on \(y = f(2x)\).A* Challenge
Q14The graph of \(y = f(x)\) passes through \((6, 4)\). Find the coordinates of the corresponding point on \(y = 2f(x - 1) + 3\).A* Challenge
Q15The graph of \(y = f(x)\) has a maximum point at \((3, 7)\). Find the coordinates of the maximum point on \(y = f(x + 2) - 5\).A* Challenge
Answers — full worked solutions
Core Skills (Q1–Q5)
Q1 — Crosses at \(x = -2, 1, 4\)
1
Three distinct (single) roots: \(x = 1\), \(x = -2\), \(x = 4\)
2
A single root always means the curve crosses straight through
Crosses the x-axis at \(x = -2, 1\) and \(4\)
Q2 — Touches at \(x = 0\), crosses at \(x = -3\)
1
\(y = x^2(x + 3)\): \(x = 0\) is a repeated (squared) root
2
\(x = -3\) is a single root
Touches the x-axis at \((0, 0)\); crosses at \((-3, 0)\)
Q3 — \(x = 0\) and \(y = 0\)
1
The curve is undefined at \(x = 0\), giving a vertical asymptote there
2
As \(x \to \pm\infty\), \(y \to 0\), giving a horizontal asymptote there
\(x = 0\) and \(y = 0\)
Q4 — \((1, 1)\) and \((2, 4)\)
1
Set equal: \(x^2 = 3x - 2 \implies x^2 - 3x + 2 = 0 \implies (x - 1)(x - 2) = 0\)
2
\(x = 1\) or \(x = 2\). Using \(y = x^2\): \(x = 1\) gives \(y = 1\); \(x = 2\) gives \(y = 4\)
\((1, 1)\) and \((2, 4)\)
Q5 — \((2, 7)\)
1
\(+4\) outside the function is a vertical translation: add 4 to the y-coordinate
\((2, 7)\)
Exam-Style (Q6–Q10)
Q6 — Crosses at \((-2, 0)\), touches at \((1, 0)\)
1
\((x + 2)\) is a single factor: crosses at \(x = -2\)
2
\((x - 1)^2\) is a repeated factor: touches at \(x = 1\)
Crosses at \((-2, 0)\); touches at \((1, 0)\)
Q7 — \(x = 0\), \(y = 0\); undefined at \(x = 0\)
1
As with \(\dfrac{1}{x}\), the curve is undefined at \(x = 0\) (vertical asymptote) and approaches 0 as \(x \to \pm\infty\) (horizontal asymptote)
2
The y-intercept would require substituting \(x = 0\), but \(1/0^2\) is undefined — so there is no y-intercept
Asymptotes \(x = 0\) and \(y = 0\); no y-intercept, since \(x = 0\) is not in the domain
Q8 — \((-1, -3)\) and \((3, 5)\)
1
Set equal: \(x^2 - 4 = 2x - 1 \implies x^2 - 2x - 3 = 0 \implies (x - 3)(x + 1) = 0\)
2
\(x = 3\) or \(x = -1\). Using \(y = 2x - 1\): \(x = 3\) gives \(y = 5\); \(x = -1\) gives \(y = -3\)
\((-1, -3)\) and \((3, 5)\)
Q9 — \((8, -2)\)
1
\(x - 5\) inside the function translates 5 units right: add 5 to the x-coordinate
\((8, -2)\)
Q10 — \((4, 18)\)
1
3 outside the function stretches vertically by factor 3: multiply the y-coordinate by 3
\((4, 18)\)
A* Challenge (Q11–Q15)
Q11 — y-intercept \((0, 12)\)
1
\((x - 2)^2\) touches at \(x = 2\); \((x + 3)\) crosses at \(x = -3\)
2
y-intercept: substitute \(x = 0\): \(y = (-2)^2(3) = 4 \times 3 = 12\)
Touches \((2, 0)\); crosses \((-3, 0)\); y-intercept \((0, 12)\)
Q12 — \((0, -3)\) and \((3, 0)\)
1
Set equal: \(x^2 - 2x - 3 = x - 3 \implies x^2 - 3x = 0 \implies x(x - 3) = 0\)
2
\(x = 0\) or \(x = 3\). Using \(y = x - 3\): \(x = 0\) gives \(y = -3\); \(x = 3\) gives \(y = 0\)
\((0, -3)\) and \((3, 0)\)
Q13 — \((1, 5)\)
1
\(y = f(2x)\) compresses horizontally by factor \(\tfrac{1}{2}\): divide the x-coordinate by 2
2
\((2, 5) \to (2 \div 2, 5)\)
\((1, 5)\)
Q14 — \((7, 11)\)
1
\(x - 1\) inside translates 1 unit right: new \(x = 6 + 1 = 7\)
2
\(2f(...)\) stretches the y-value by 2: \(2 \times 4 = 8\)
3
\(+3\) outside translates up by 3: \(8 + 3 = 11\)
\((7, 11)\)
Q15 — \((1, 2)\)
1
\(x + 2\) inside translates 2 units left: new \(x = 3 - 2 = 1\)
2
\(-5\) outside translates 5 units down: new \(y = 7 - 5 = 2\)
3
Translations preserve the maximum/minimum nature of a point, so this is still a maximum
Maximum point \((1, 2)\)
Summary
- A single root means a cubic crosses the x-axis there; a repeated (squared) root means it touches and turns back.
- \(y = \dfrac{1}{x}\) and \(y = \dfrac{1}{x^2}\) never touch either axis — \(x = 0\) and \(y = 0\) are both asymptotes.
- Find points of intersection by setting the two expressions for y equal, solving for x, then substituting back for y.
- For combined transformations, apply the x-transformation (inside the brackets) first, then the transformation to the whole function (outside the brackets).
- Translating a graph never changes whether a point is a maximum or minimum — only its coordinates.
These sketching and transformation skills reuse the completing-the-square and simultaneous equation techniques from earlier in Pure Year 1, and feed forward into Straight Line Graphs, where the same intersection technique finds triangle vertices.
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