The same-looking curve may follow a different rule
Imagine two curves displayed on the same set of axes. Both lie above the horizontal axis, both rise sharply near the vertical axis and both become low as the horizontal distance grows. One is y = 12/x. The other is y = 12/x². At x = 1 they have the same value, 12. That single point cannot identify which rule produced the curve. At x = 2, however, the first gives 6 and the second gives 3. The difference is not an artistic detail in the drawing. It is a mathematical consequence of an exponent.
A G3 SEC Mathematics question may ask you to select a matching graph, calculate a missing coordinate, explain why a claimed pattern is wrong or interpret a power-function model in context. The most dependable approach begins with the algebra. Does doubling x double y, square its effect, cube its effect, halve y or reduce y to one quarter? What happens when x is negative? Is x = 0 allowed? A graph becomes easier to understand when each visible feature follows from a simple tested value.
This guide focuses on power-function graphs and their combinations, moving from familiar straight lines and quadratics toward reciprocals, reciprocal squares and cubic curves. It deliberately does not repeat the separate questions owned by earlier volumes on exponential growth or general graph-family choice. The goal is a method that survives when a question changes the coefficient, sign, scale or description of the axes.
The official 2027 SEC Mathematics framework
The Singapore–Cambridge Secondary Education Certificate begins in 2027. G3 Mathematics uses syllabus K310. According to the official 2027 SEAB Mathematics K310 syllabus, the Functions and Graphs content includes power functions of the form y = axⁿ with n = −2, −1, 0, 1, 2 and 3, alongside simple sums of power functions within the specified scope. These are the exponents used in this guide; the exercises are original examples, not official examination questions.
For K310, Paper 1 lasts two hours and fifteen minutes and carries 90 marks, or 50% of the subject’s assessment. Paper 2 also lasts two hours and fifteen minutes, carries 90 marks and 50%, and includes a real-world problem-solving question. Unless a question specifies otherwise, non-exact numerical answers are generally given to three significant figures and angles to one decimal place. The 2027 SEAB G3 syllabus directory is the authority for the actual subject entry and paper specifications.
A G3 question does not become harder simply because its formula contains a negative exponent. The student’s job remains to substitute values accurately, apply algebraic laws, identify the relevant restrictions and justify the conclusion requested. Do not introduce calculus or other advanced methods merely to make an ordinary graph question look more sophisticated. Often a table, sign test or one correctly identified coordinate is sufficient.
Six powers, six basic behaviours
For n = 1, y = ax gives a straight line through the origin. When a is positive, y grows with x; when a is negative, the line slopes downward. For n = 2, y = ax² is a parabola symmetric about the vertical axis: x and −x produce equal outputs. With positive a it opens upwards, and with negative a it opens downwards. For n = 3, y = ax³ takes opposite signs for opposite nonzero x when a is positive, because a cube preserves the sign of its base.
For n = −1, y = ax⁻¹ means y = a/x. Dividing by x requires x ≠ 0. For positive a, a positive x gives a positive y and a negative x gives a negative y. As the magnitude of x increases, the magnitude of y decreases. For n = −2, y = ax⁻² means y = a/x²; x ≠ 0 again. Squaring makes positive and negative inputs produce the same denominator, so a positive coefficient gives positive outputs on both sides of zero.
For n = 0, x⁰ = 1 for nonzero x, giving a constant power-function value y = a. At x = 0, the expression 0⁰ requires care and should not be handled by blindly substituting into the exponent law. In ordinary graph interpretation, the essential skill is recognising the constant rule and following the domain stated in the question. Do not pretend that an algebraic convention settles every possible representation without context.
A coefficient a scales the values and can reverse their sign. Compare y = 2/x² with y = −2/x². Their magnitudes agree at each permitted input, but one lies above the x-axis and the other below. Compare y = 2x³ with y = −2x³. One gives a positive output at positive x; the other gives a negative output. A change of sign can change a graph’s orientation without changing its exponent.
A reliable inspection order
Start with the variable’s domain. If the rule has x in the denominator, x = 0 is not in its domain. An absent graph point at zero is not a mysterious plotting error. It is a mathematical restriction. If the question states that x represents distance, time or number of items, the physical domain may be narrower than the algebraic one. Negative distances or fractional people may not be valid interpretations even when the formula can be evaluated.
Next test sign and symmetry. Does changing x to −x leave y unchanged? That is true for even powers such as x², x⁻² and the constant x⁰ when defined. Does the sign of y reverse? That happens for odd powers such as x and x³, and for x⁻¹ with nonzero x. A two-point test at x = 2 and x = −2 often distinguishes reciprocal from reciprocal-square behaviour immediately.
Then compare magnitude. When x doubles from 2 to 4, y = ax changes by a factor of two, y = ax² by four, and y = ax³ by eight. Over the same change, y = a/x changes by one half and y = a/x² by one quarter, provided the inputs are nonzero. That multiplication rule is a practical way to identify a power from a table or a short word problem.
Finally check intercepts and limiting behaviour without overclaiming. For positive n and a ≠ 0, y = axⁿ passes through (0,0). Reciprocal and reciprocal-square graphs are not defined at x = 0 and do not intersect the x-axis for finite permitted x when a ≠ 0. Their plotted values can approach zero as |x| grows, but approaching a line is not the same as reaching it. A careful rough sketch expresses these facts without inventing a point where the formula is undefined.
Power functions are not exponential functions
y = 3x² is a power function: the variable is the base, and the exponent 2 is fixed. y = 3²ˣ, or a more ordinary example y = 2ˣ, is exponential: the variable appears in the exponent. A question that says the output doubles every time x increases by one may suggest an exponential pattern rather than one of the six basic power forms. Do not decide from the word ‘growth’ alone. Examine what changes and what remains fixed.
Likewise y = 4/x is not a declining straight line merely because its values get smaller as x grows. At x = 1, 2 and 4, its values are 4, 2 and 1. The absolute decreases are 2 and then 1, not a fixed difference, while the factor when doubling x is consistently one half. That is multiplicative inverse behaviour rather than constant additive change.
When the expression contains a sum such as y = x³ − 2x + 1, there is no single exponent controlling all the values. Evaluate each term and combine them with the proper signs. A graph’s appearance can be more complex than the individual building blocks. The syllabus includes appropriate simple sums, but the same safe method remains: find the domain and compute reliable points before drawing conclusions.
Thirty-two original worked graph decisions
The worked cases vary the exponent, sign, coefficient, permitted inputs and context. Solve each without reading the worked answer first. If the result seems odd, substitute it back into the given rule. Build a small table where useful; the point is not to memorise a shape but to explain it from algebra.
For graph sketches, do not needlessly draw the curves with perfect artistic precision. Label key coordinates and axes, show the correct quadrants and identify excluded values. When a context limits x to positive quantities, sketch and interpret only the meaningful part of the curve rather than discussing negative values as if they were physically observed.
Power-graph case 01: One input, two candidates
Question. A plotted point is (1,12). Could it distinguish y = 12/x from y = 12/x²?
Worked answer. No. Substituting x = 1 gives 12 for both rules. A second nonzero input other than ±1 is needed. For example x = 2 gives 6 for the reciprocal but 3 for the reciprocal square.
Why it matters. A model must be distinguished by evidence that its competitors do not share. One convenient matching point rarely identifies a whole graph.
Power-graph case 02: Doubling x
Question. For y = 18/x, compare y at x = 3 and x = 6.
Worked answer. At x = 3, y = 6; at x = 6, y = 3. Doubling the positive input halves the output. The product xy remains 18 for every permitted x.
Why it matters. A fixed product is a quick inverse-proportion check. It is not a fixed difference, so do not treat the graph as a straight decreasing line.
Power-graph case 03: Quartering the output
Question. For y = 72/x², calculate y at x = 3 and x = 6.
Worked answer. At x = 3, y = 72/9 = 8. At x = 6, y = 72/36 = 2. Doubling x reduces y to one quarter of its earlier value.
Why it matters. The square applies to the whole denominator x². Simply halving y would confuse inverse-square variation with inverse variation.
Power-graph case 04: Positive and negative inputs
Question. For y = 20/x, evaluate y at x = 4 and x = −4.
Worked answer. The outputs are 5 and −5. Changing the sign of x changes the sign of y when the coefficient is positive.
Why it matters. This reciprocal graph has branches in opposite-sign input-output regions. The algebra, not a remembered curve drawing, establishes their placement.
Power-graph case 05: Equal negative and positive inputs
Question. For y = 20/x², evaluate at x = 4 and −4.
Worked answer. Both give 20/16 = 1.25. The squared denominator is positive at either nonzero input, so the outputs are equal.
Why it matters. The inverse-square shape is symmetric about the vertical axis. It does not become negative merely because x is negative.
Power-graph case 06: A negative coefficient
Question. Sketch the sign of y = −9/x² for x = −3, −1, 1 and 3.
Worked answer. The outputs are −1, −9, −9 and −1. Both branches lie below the horizontal axis, and x = 0 is excluded.
Why it matters. Do not confuse the coefficient’s negative sign with the exponent’s negative sign. An exponent of −2 means divide by x²; the coefficient determines whether the resulting value is positive or negative.
Power-graph case 07: An undefined reading
Question. A student substitutes x = 0 into y = 6/x and records y = 0. What is wrong?
Worked answer. Division by zero is undefined. There is no valid point on this function at x = 0. The output does not become zero just because the denominator is zero.
Why it matters. A missing point is not a plot error to be patched with an invented value. State the domain restriction before describing the graph.
Power-graph case 08: Approaching but never reaching
Question. For y = 5/x² with x positive, what happens to y as x becomes very large?
Worked answer. The positive value becomes smaller and approaches zero, but does not equal zero for any finite positive x. At x = 10, y = 0.05; at x = 100, y = 0.0005.
Why it matters. ‘Approaches zero’ is mathematically different from ‘becomes zero’. Avoid drawing a false intercept merely because a curve seems to touch the axis on a small sketch.
Power-graph case 09: Growth of a cube
Question. For y = 2x³, compare y at x = 2 and x = 4.
Worked answer. At x = 2, y = 16. At x = 4, y = 128, which is eight times 16. Doubling x multiplies x³ by 2³ = 8.
Why it matters. A cubic power does not behave like a quadratic, which would multiply the output by four under doubling. The exponent controls the factor.
Power-graph case 10: A cubic negative input
Question. For y = 3x³, evaluate y at x = −2 and x = 2.
Worked answer. The results are −24 and 24 respectively. Cubing a negative number produces a negative value, while cubing the corresponding positive number produces a positive value.
Why it matters. A cubic power has odd symmetry for a single term. Check sign before multiplying the coefficient.
Power-graph case 11: Straight line or parabola?
Question. A table gives (1,4), (2,8), (3,12). Is y = 4x or y = 4x² the better match?
Worked answer. y = 4x matches all three: 4, 8 and 12. y = 4x² would give 4, 16 and 36, so it fails the second and third readings.
Why it matters. A model is supported by agreement across distinct inputs. An isolated first point is insufficient for choosing the exponent.
Power-graph case 12: A squared pattern
Question. A table has x = 1, 2, 3 and y = 3, 12, 27. Identify a basic power rule.
Worked answer. The outputs match y = 3x²: 3, 3×4 = 12 and 3×9 = 27. The input change from 1 to 2 multiplies y by four, and from 1 to 3 by nine.
Why it matters. The values do not increase by a constant amount, so the graph is not a straight line. Multiplicative comparisons reveal the squared structure.
Power-graph case 13: A reciprocal table
Question. Given x = 1, 2, 4 and y = 16, 8, 4, find a simple rule.
Worked answer. The pairs have a constant product xy = 16, so y = 16/x fits the data. Substituting any provided nonzero x confirms the rule.
Why it matters. The graph is a reciprocal relationship because the output halves when the input doubles. Verify the constant product before naming the family.
Power-graph case 14: Inverse-square table
Question. Given x = 1, 2, 4 and y = 48, 12, 3, identify the rule.
Worked answer. The values are consistent with y = 48/x². Doubling x divides y by four each time. For example 48/2² = 12 and 48/4² = 3.
Why it matters. A student who assumes y = a/x would predict 24 at x = 2 and be contradicted immediately. Test the exponent, not just the direction of change.
Power-graph case 15: Coefficient from one point
Question. A curve has y = a/x² and passes through (3,4). Find a.
Worked answer. Substitute 4 = a/9, so a = 36. The full rule is y = 36/x² for nonzero x.
Why it matters. A single point can determine a coefficient once the exponent and form are given. It cannot determine both form and coefficient without additional evidence.
Power-graph case 16: Coefficient with a negative input
Question. The graph y = a/x passes through (−5,3). Find a.
Worked answer. Substitute 3 = a/(−5), hence a = −15. The rule y = −15/x gives a positive output at x = −5, as required.
Why it matters. A negative product is not an arithmetic accident. Check the sign of a by substituting the point back into the rule.
Power-graph case 17: A negative squared value?
Question. Can y = 7/x² equal −2 for any real nonzero x?
Worked answer. No. x² is positive for any nonzero real x, and 7/x² is therefore positive. It cannot equal a negative number.
Why it matters. A sign argument can rule out a solution without elaborate algebra. First inspect what values a function is capable of producing.
Power-graph case 18: A reciprocal equation
Question. Solve 12/x = 3 for real x.
Worked answer. Since x ≠ 0, multiply both sides by x: 12 = 3x, so x = 4. Check: 12/4 = 3.
Why it matters. Clearing denominators is allowed only when you preserve the restriction that the denominator cannot be zero. The final answer satisfies it.
Power-graph case 19: An inverse-square equation
Question. Solve 36/x² = 4 for real x.
Worked answer. Multiply by x², which is nonzero: 36 = 4x², so x² = 9 and x = ±3. Both values satisfy the original equation.
Why it matters. Squaring makes opposite inputs share an output. Forgetting the negative root is a common lost-mark error.
Power-graph case 20: Not every power has a negative branch
Question. A learner says y = x² has a negative output when x is negative. Correct the claim.
Worked answer. For x = −3, x² = 9, not −9. Squaring a negative number gives a positive number. With a positive coefficient, y = ax² is nonnegative for all real x.
Why it matters. The order of operations matters. (−3)² = 9, while −3² conventionally means −(3²) = −9 unless brackets change the expression.
Power-graph case 21: Sign of a cubic coefficient
Question. For y = −2x³, find y at x = −2.
Worked answer. x³ = −8, so −2×(−8) = 16. The graph gives positive y at that negative x.
Why it matters. There are two negative signs from different sources: one in x³ and one in the coefficient. Evaluate the power first, then multiply.
Power-graph case 22: Constant power
Question. For nonzero x, what does y = 7x⁰ become?
Worked answer. Since x⁰ = 1 for nonzero x, y = 7. The graph has constant output over the stated domain. The exponent zero does not make the result zero.
Why it matters. If a question includes x = 0, treat the expression and its declared domain with care; do not resolve 0⁰ by careless substitution.
Power-graph case 23: Comparing two small inputs
Question. Compare y = 1/x and y = 1/x² at x = 1/2.
Worked answer. For 1/x, the output is 2; for 1/x², it is 4. Both become larger than 1 for this positive input below 1, but the inverse-square value is larger.
Why it matters. Negative exponents can make small positive inputs produce large outputs. Do not generalise from the x greater than 1 region alone.
Power-graph case 24: Which curve is even?
Question. State whether y = 4/x, y = 4/x² and y = 4x³ have equal outputs at x and −x.
Worked answer. Only y = 4/x² has equal outputs at x and −x for every permitted nonzero input. The other two reverse sign when x changes sign.
Why it matters. This is a symmetry test, not a claim about which curve is ‘better’. Check the exponent’s parity and the domain.
Power-graph case 25: Practical domain
Question. A simple model states that a measured quantity is y = 120/x, where x is the number of identical active machines. Does x = −3 have a physical interpretation?
Worked answer. No. Although the algebraic rule can be evaluated at −3, a negative number of machines has no physical meaning here. In a realistic discrete model, x must be a positive whole number.
Why it matters. Mathematical domain and real-world domain are different. A correct graph can still be interpreted incorrectly if impossible inputs are treated as actual cases.
Power-graph case 26: The graph at two scales
Question. A sketch makes y = 2/x² look nearly flat for x from 20 to 50. Has the rule become constant?
Worked answer. No. The values continue to change: at x = 20, y = 0.005; at x = 50, y = 0.0008. They only appear close to zero because of the graph’s scale.
Why it matters. Visible flatness is not an algebraic identity. Test two values and read the axis resolution before declaring that a quantity is unchanged.
Power-graph case 27: Simple sum of powers
Question. Evaluate y = x³ − 2x + 1 at x = 2.
Worked answer. Substitute term by term: 2³ − 2(2) + 1 = 8 − 4 + 1 = 5. The expression contains three power terms: cubic, linear and constant.
Why it matters. A combined graph may not look exactly like any one basic graph. Substitution gives a secure coordinate without trying to guess the whole shape.
Power-graph case 28: Three zeroes in a sum
Question. Solve x³ − x = 0.
Worked answer. Factor x(x²−1) = x(x−1)(x+1) = 0, giving x = −1, 0 and 1. All three are valid x-intercepts of the corresponding combined function.
Why it matters. When terms are added, a graph can cross the horizontal axis more than once. Do not apply the single-term ‘only zero’ expectation to a sum.
Power-graph case 29: Combining a reciprocal and constant
Question. For y = 8/x + 2, calculate y when x = 4 and state the excluded input.
Worked answer. At x = 4, y = 2 + 2 = 4. The expression is undefined when x = 0 because the reciprocal term divides by x. Adding 2 does not remove the restriction.
Why it matters. Every term in a sum must be defined. One invalid term makes the whole expression invalid at that input.
Power-graph case 30: An x-squared plus constant
Question. Given y = −x² + 9, find its x-intercepts.
Worked answer. Set y = 0: −x² + 9 = 0, so x² = 9 and x = ±3. The graph meets the horizontal axis at (−3,0) and (3,0).
Why it matters. An even power with a constant shift may have two intercepts instead of only the origin. Work from the full expression, not a remembered basic sketch.
Power-graph case 31: Average versus instantaneous slope
Question. A question asks for average change in y as x rises from 1 to 3 for y = x². Find it.
Worked answer. The endpoints are y(1) = 1 and y(3) = 9. Average change per unit x is (9−1)/(3−1) = 4. This is a chord gradient over an interval, not necessarily the local gradient at a point.
Why it matters. SEC graph reading can use simple differences without calculus. Name whether the result describes an interval or one location on the curve.
Power-graph case 32: An exponential lookalike
Question. A table shows y doubles each time x increases by 1: at x = 1,2,3,4 the values are 3,6,12,24. Is it y = ax²?
Worked answer. No. For a quadratic term ax², the values at x = 1 and 2 would differ by a factor of four, not two. The repeated-doubling pattern is characteristic of an exponential relationship such as y = 1.5×2ˣ.
Why it matters. Growth alone does not make a power function. Check whether the variable is raised to a fixed exponent or appears in the exponent itself.
Power-graph case 33: A final model check
Question. A pupil claims y = a/x² fits data (2,10) and (4,5). Can one a work for both points?
Worked answer. The first point requires a = 10×2² = 40; the second requires a = 5×4² = 80. Because the implied coefficients disagree, no single rule of that form fits both points exactly.
Why it matters. Always test all supplied points against the candidate model. An answer is not established merely because one point can be made to fit.
Longer integrated example: choosing the curve from a table
A school introduces a fictional quantitative model for how a measured signal changes with distance. The table shows the signal has values 45, 11.25 and 5 when distance is 2, 4 and 6 units. You are asked to choose between a straight line, y = a/x, y = a/x² and a cubic model. First ask whether the signal behaves as a fixed ratio under a doubling of distance. The distance changes from 2 to 4 while the signal changes from 45 to 11.25, exactly one quarter. That strongly suggests inverse-square behaviour, but one matching ratio alone should still be checked against another point.
Compute the constant implied by y = a/x². At x = 2, a = 45×4 = 180. At x = 4, a = 11.25×16 = 180. At x = 6, a = 5×36 = 180. The same a works for all three values, so y = 180/x² fits the supplied table exactly. State that this is a mathematical fit to the fictional measurements; do not claim the table proves a universal physical law outside the stipulated situation.
Now predict the output at x = 3: y = 180/9 = 20. Interpret the graph only over the stated meaningful distance range. If x represents a positive distance, negative values have no physical role even though the algebraic graph can include them. If the source of the signal has a finite size or the instrument saturates near very short distances, the model may fail outside its measured range. Predicting is not the same as validating the model over every possible input.
A second learner argues that the function should be y = 90/x because at x = 2 it also gives 45. That model fails at x = 4: it predicts 22.5 rather than the observed 11.25. The conflict is settled by evidence, not by which graph looks more familiar. A powerful examination habit is to determine a coefficient from one point, then use a different point as a test. The second point can falsify the proposed form.
The question finally asks for the distance at which the model predicts a signal of 7.2. Solve 180/x² = 7.2, so x² = 25 and x = ±5 mathematically. Because the context defines positive distance, x = 5 is the physically meaningful answer. Keep both steps in working: the equation has two real roots, but the model’s domain determines which one can be used. This is what it means to combine algebra with real-world interpretation.
How to check a sketched power graph
First locate a point that you can calculate without difficulty, such as x = 1. For y = axⁿ, that output equals a for every listed integer exponent because 1 raised to a fixed integer power is 1. This gives the point (1,a) whenever the input is permitted. It is a fast coefficient and vertical-scale check, but it does not distinguish the exponent by itself.
Second inspect a positive input greater than one, often x = 2. For n = −2 the factor is one quarter, for −1 it is one half, for 0 it is one, for 1 it is two, for 2 it is four and for 3 it is eight, relative to y(1) = a when a is nonzero. These six factors provide a compact mental table. You do not need an elaborate drawing to reject several mismatching graph choices.
Third try a negative input if the algebraic domain allows it. Even powers preserve the output when the sign of x changes; odd powers reverse it. Then decide whether the curve can pass through x = 0 and whether it can cross the horizontal axis. The signs, symmetry, intercepts and domain are independent checks. A sketch that passes only one check is not necessarily correct.
Finally translate the question’s words back into the mathematics. ‘Doubles when the input doubles’ suggests a power-one relation in the relevant setting. ‘Becomes one quarter when distance doubles’ suggests a negative-square power. ‘Doubles at every equal step in time’ suggests a different, exponential class of model. State the reason rather than relying on a visual memory of the curve.
A practical revision route for a new graph family
On the first session, write all six exponent values and calculate outputs for x = 1, 2 and −2 where permitted. Explain each sign and each excluded input aloud. On the second session, draw four small graphs from calculated points: positive reciprocal, positive reciprocal square, a quadratic and a cubic. Put each graph’s rule directly beside it so the picture and algebra reinforce one another.
On the third session, practise identifying a model from three points. Use one point to calculate the coefficient and the remaining points to test whether the exponent is right. On the fourth session, mix graphs from earlier learning: linear, quadratic, cubic, reciprocal, reciprocal square and exponential. Force yourself to state a numerical test for every choice, rather than saying that one curve merely ‘looks right’.
On the fifth session, solve real-world questions in which the domain matters. Write the algebraic solutions first, then explain why a particular solution is or is not sensible for time, length, counts or other quantities. On the sixth session, attempt simple sums of power functions and calculate coordinates carefully. On the seventh session, take an unfamiliar teacher-approved graph question and solve it without referring to memorised case numbers.
After marking, label the earliest source of any error: exponent law, substitution, negative number, domain, model choice, plotting scale or contextual interpretation. A student who repeatedly misses the negative branch of a cubic needs a sign check, not more pages of random revision. The next attempt should target the cause directly. This is how practice becomes progress.
Where this guide fits in the English–Mathematics–Science route
The exponential-function family has a separate owner: Volume 0226. For choosing among common model families from data, use Volume 0230. For tangent gradients on a curved graph rather than power-function shape, use Volume 0222. These are distinct tasks; this volume’s own focus is the exponent, sign, domain and graph of a power function.
The immediate previous learning sequence includes Volume 0266 on possibility diagrams for systematic mathematical representation, Volume 0267 on reaction-rate curves for interpreting a measured rate and Volume 0268 for checking claims across subjects. Each develops a different aspect of disciplined problem solving rather than copying the same question.
For earlier mathematical confidence, return to the PSLE Learning Guide and the Mathematics Learning Hub. For 2027 syllabus scope and examination arrangements, use the official G3 Mathematics K310 document. The examples here are original practice exercises. The transferable habit is to test a model with numbers, name its restrictions and let the algebra explain the picture.
