Edexcel IAL P1 Graphs and Transformations: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

P1 graphs means sketching cubics, quartics and reciprocal graphs from their factors and asymptotes, using intersections to count the solutions of an equation, and transforming $y = f (x)$ . A sketch needs the right shape, the axis intercepts labelled and any asymptotes drawn. Changes inside the bracket act on $x$ and work "backwards".

P1 · Pure Mathematics 1
Calculator allowed
3 worked examples
30 practice questions
Tested in 4 of the last 4 papers

Practise P1 graphs and transformations →

Notes
Formulas
Worked examples
Mistakes
FAQ

P1 Graphs and Transformations notes

Sketching cubics and quartics

Factorise, mark the roots on the $x$ -axis, find the $y$ -intercept, and decide the shape from the sign of the highest power. $y = (x - 1) (x + 2) (x - 3)$ crosses the $x$ -axis at $- 2$ , $1$ and $3$ and the $y$ -axis at $6$ ; with a positive $x^{3}$ term it rises from bottom left to top right. A squared factor is a repeated root, where the curve touches the axis: $y = x^{2} (x - 3)$ touches at the origin and crosses at $x = 3$ .

Reciprocal graphs

$y = \frac{k}{x}$ has asymptotes $x = 0$ and $y = 0$ ; for $k > 0$ its branches lie in the first and third quadrants. $y = \frac{k}{x^{2}}$ has the same asymptotes, but for $k > 0$ both branches are above the $x$ -axis.

Intersections and the number of solutions

The solutions of $f (x) = g (x)$ are the $x$ -coordinates where $y = f (x)$ and $y = g (x)$ meet, so a sketch tells you how many there are. To show a line and a curve never meet, set them equal and show the resulting quadratic has a negative discriminant.

Transformations of graphs

$y = f (x) + a$ moves the graph up by $a$ ; $y = f (x + a)$ moves it left by $a$ ; $y = a f (x)$ stretches it vertically by factor $a$ ; $y = f (a x)$ stretches it horizontally by factor $\frac{1}{a}$ ; $y = - f (x)$ and $y = f (- x)$ reflect it in the $x$ - and $y$ -axes. If $y = f (x)$ has a maximum at $(2, 5)$ , then $y = f (x - 3)$ has it at $(5, 5)$ , $y = 2 f (x)$ at $(2, 10)$ , $y = f (2 x)$ at $(1, 5)$ and $y = - f (x)$ has a minimum at $(2, - 5)$ .

Key formulas

Transformations: $f (x) + a$ up $a$ ; $f (x + a)$ left $a$ ; $a f (x)$ vertical × $a$ ; $f (a x)$ horizontal × $\frac{1}{a}$ Learn it

In the formula booklet = printed in Pearson's IAL formulae booklet, so know where to find it. Learn it = not printed there, so learn it by heart. All P1 formulas on one page (free PDF) →

P1 Graphs and Transformations worked examples

Each mark scheme is checked line by line before it is published here. M marks are for method, A marks for accurate answers that follow from them, B marks for independent results.

Worked example 1 · 4 marks · hard

A cubic curve crosses the x-axis at $x = - 2$ , $x = 1$ , and $x = 4$ . It crosses the y-axis at the point $(0, 16)$ . Find the equation of the curve in the form $y = a x^{3} + b x^{2} + c x + d$ .

Mark scheme

M1 for writing the equation in the factored form $y = k (x + 2) (x - 1) (x - 4)$ .
M1 for substituting $(0, 16)$ to find $k$ : $16 = k (2) (- 1) (- 4) ⟹ 16 = 8 k$ .
A1 for finding $k = 2$ .
A1 for expanding to get $y = 2 (x^{3} - 3 x^{2} - 6 x + 8) = 2 x^{3} - 6 x^{2} - 12 x + 16$ .

4 steps · 1 to fill in

Worked example 2 · 4 marks · hard

Find the x-coordinates of all the points of intersection between the curves $y = x (x + 3)$ and $y = x^{3} (x + 3)$ .

Mark scheme

M1 for equating the functions: $x^{3} (x + 3) = x (x + 3)$ .
M1 for bringing all terms to one side and factorising: $x (x + 3) (x^{2} - 1) = 0$ .
M1 for further factorising $x^{2} - 1$ to $(x - 1) (x + 1)$ .
A1 for listing all four correct x-coordinates: $x = 0, - 3, 1, - 1$ .

4 steps

Worked example 3 · 4 marks · hard

Show that the line $y = 2 x - 7$ does not intersect the curve $y = x^{2} - 3 x + 5$ .

Mark scheme

M1 for equating the line and the curve: $x^{2} - 3 x + 5 = 2 x - 7$ .
M1 for rearranging into a quadratic equation: $x^{2} - 5 x + 12 = 0$ .
M1 for calculating the discriminant: $b^{2} - 4 a c = (- 5)^{2} - 4 (1) (12) = 25 - 48 = - 23$ .
A1 for concluding that since the discriminant is less than zero, there are no real roots, and thus no points of intersection.

4 steps · 1 to fill in

Play with it Translate a parabola: drag the sliders

The dashed curve is y = f(x), where f(x) = x². Change h and k so that y = f(x − h) + k passes through all the points. Write the translation as a vector.

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Common P1 graphs and transformations mistakes

Moving the wrong way. $y = f (x + 3)$ moves the graph 3 to the left, not right.
Unlabelled intercepts. A sketch earns its marks for the points where it meets the axes: write their coordinates on the diagram.
Crossing at a repeated root. At a squared factor the curve touches the x-axis and turns back; it does not cross.
Asymptotes left off. Draw and label asymptotes as dashed lines with their equations, such as $x = 0$ and $y = 0$ .

Every P1 mistake from the examiner reports, with the paper it came from →

On your calculator: how to find intersections, zeros, maximum and minimum points. A calculator is allowed in every IAL unit.

P1 Graphs and Transformations FAQ

How do I sketch a cubic in IAL P1?

Factorise it, mark the roots on the x-axis, find where it crosses the y-axis, and use the sign of the x³ term for the overall shape: positive rises from bottom left to top right, negative falls. A repeated root is a point where the curve touches the axis.

Why does f(x + a) move the graph left?

Because the same y-value is now reached when x is a smaller by a: the point that was at x = 2 on y = f(x) is at x = 2 − a on y = f(x + a).

Is a calculator allowed in IAL P1?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P1 included. Method marks are for the working you write down, so show it.

How long is the IAL P1 exam?

Each IAL maths unit, P1 included, is one written paper of 1 hour 30 minutes worth 75 marks.

How often does graphs and transformations come up in P1?

In our mapping of the 4 Pearson P1 papers from June 2024 to June 2025, graphs and transformations was tested in 4 of them (7 questions in all). Any topic can be examined in any series.

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More P1: Algebra · Coordinate Geometry · Trigonometry · Differentiation · Integration

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Other units: IAL P2 · IAL P3 · IAL P4 · IAL S1 · IAL M1

Edexcel IAL P1 Graphs and Transformations: exam questions and answers, notes and worked examples

Try it first

P1 Graphs and Transformations notes

Sketching cubics and quartics

Reciprocal graphs

Intersections and the number of solutions

Transformations of graphs

Key formulas

P1 Graphs and Transformations worked examples

Worked example 1 · 4 marks · hard

Mark scheme

Worked example 2 · 4 marks · hard

Mark scheme

Worked example 3 · 4 marks · hard

Mark scheme

Common P1 graphs and transformations mistakes

P1 Graphs and Transformations FAQ

How do I sketch a cubic in IAL P1?

Why does f(x + a) move the graph left?

Is a calculator allowed in IAL P1?

How long is the IAL P1 exam?

How often does graphs and transformations come up in P1?

Next steps

Your P1 revision kit