A Level · Cambridge (CIE) · 9709

Translate, stretch and reflect a graph vertically

Track how changing a function's output moves points while leaving their input coordinates unchanged.

Section 1.2 · Functions

Pure Mathematics 1 · Functions · 1.2

Pure Mathematics 1 (Paper 1) · AS & A Level Mathematics 9709, 2026–2027. Written practice is self-checked; show your reasoning and domain restrictions.

Before you begin

Identify the coordinates of a point on a function graph.
Substitute a value into a formula and multiply signed numbers.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

A point (1, 4) lies on y = f(x). What does f(1) equal?

What you will learn

Describe y = f(x) + a and y = af(x) using translations, stretches and reflections.
Apply simple output transformations in the specified order.

Change the output, keep the input

If (u, v) lies on y = f(x), then (u, v + a) lies on y = f(x) + a. This is a translation by vector (0, a): upward for a > 0, downward for a < 0. The input coordinate and domain do not change. The range translates by the same amount.

For y = af(x), points become (u, av). If a > 0, this is a stretch parallel to the y-axis with factor a; a factor between 0 and 1 reduces vertical distances from the x-axis. If a < 0, reflect in the x-axis and stretch parallel to the y-axis with factor |a|. The special value a = 0 makes a constant zero function on the original domain and loses information about its original range.

Use the final rule to check transformation order

For y = 2f(x) + 3, first multiply each original output by 2, then add 3. A point (−1, 2) becomes (−1, 7). This is a vertical stretch with factor 2 followed by an upward translation of 3. Reversing those operations would give 2(f(x) + 3) = 2f(x) + 6 instead.

If f has minimum −4 at x = 2 and range y ≥ −4, then y = −3f(x) + 2 has maximum 14 at the same input. The negative multiplier reverses the output inequality: −3f(x) ≤ 12, so the new range is y ≤ 14. A minimum can become a maximum when the graph is reflected.

Pause and explain

What does y = −f(x) do to the original graph?

Put the idea to work

Worked example

Describe y = 2f(x) + 3 from y = f(x), and find the image of the point (−1, 2).

Show the worked solution

Multiply output values by 2: stretch parallel to the y-axis with factor 2.
Then add 3 to outputs: translate upward by 3.
Keep the input −1; the new output is 2(2) + 3 = 7.

Answer Vertical stretch factor 2, then translation (0, 3); point (−1, 7).

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

A graph y = f(x) contains (1, 4). Describe y = f(x) + 5 and give the image of that point.

Give me a hint

An added constant outside f changes only the output.

Compare my reasoning

The rule adds 5 to every output.
This is translation by vector (0, 5), with the input coordinate unchanged.
The new point is (1, 4 + 5) = (1, 9).

Translation (0, 5); point (1, 9).

Look for these in your work

Name a vertical translation with its direction and amount.
Preserve x and add 5 to y.

2 · Independent

The function f has range y ≥ −4 and a minimum at (2, −4). Find the range and extreme point of y = −3f(x) + 2.

Give me a hint

Multiplication by a negative number reverses the output inequality.

Compare my reasoning

Multiply the original range inequality by −3: −3f(x) ≤ 12.
Add 2 to get the new range y ≤ 14.
The original minimum point becomes (2, −3(−4) + 2) = (2, 14), now a maximum.

Range y ≤ 14; maximum at (2, 14).

Look for these in your work

Reverse the range inequality correctly.
Keep the input coordinate and identify the new extreme type.

3 · Transfer

At input 1, f has output 3. Compare the point after a vertical stretch factor 2 followed by a downward translation of 1 with the point after the reverse order.

Give me a hint

Write an output rule for each order before calculating.

Compare my reasoning

Stretch then translate gives 2f(x) − 1, so output 2(3) − 1 = 5.
Translate then stretch gives 2(f(x) − 1), so output 2(3 − 1) = 4.
Both keep the input 1, but the different rules give different points.

Stretch then translate: (1, 5). Reverse order: (1, 4).

Look for these in your work

Express both orders with correctly placed brackets.
Use the different outputs to explain why order matters.

Common mistakes

Output transformations change y-coordinates, not x-coordinates.
Use a positive scale factor together with a named reflection when describing a negative multiplier.
Changing the order of a stretch and a translation usually changes the final graph.

Recall without your notes

What can you explain now?

The graph y = f(x) contains (3, −2). Where does this point move on y = −2f(x) + 1?

Compare with the explanation

(3, 5).

Keep x = 3 and transform its output: −2(−2) + 1 = 5.

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Make it stick

Describe f(x) + a, af(x) and −f(x) by tracking one point. Give an example where reversing a stretch and a translation changes the result.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.