A Graph Is Evidence, Not a Proof: Windows, Missing Roots and Numerical Artefacts

A graph is one of Mathematics’ most powerful representations because it makes behaviour visible. It can reveal intercepts, turning points, asymptotes, trends, symmetry and relationships between quantities in seconds. That speed is also the source of a common mistake: treating what appears on a screen as everything the function can do.

The mature rule is different. A graph is evidence. It can suggest a conjecture, locate a candidate, test plausibility and expose structure. But the displayed image is produced from a function, a domain, a viewing window, a sampling process, a plotting algorithm and a finite screen. The picture is therefore an observation of mathematical behaviour under selected conditions, not automatically a proof about all values.

1. The graphing question comes before the graph

Before opening a graphing calculator or digital plotter, state the mathematical job. Typical jobs include:

• locate approximate roots;
• estimate a turning point;
• compare two functions;
• inspect long-run behaviour;
• test whether an algebraic answer is plausible;
• identify an interval for a numerical method;
• visualise a transformation;
• look for a counterexample to a claim.

The chosen job determines what window and what verification are appropriate. A window suitable for seeing a local turning point may be useless for deciding how many real roots a polynomial has.

2. The viewing window is part of the claim

Suppose a learner graphs y=x-50 using a window from x=-10 to x=10. The screen shows no x-intercept. It would be false to conclude that the function has no root. The root is x=50, simply outside the selected window.

Every statement based on a graph should therefore be mentally qualified: “In this displayed window, I observe…” That phrase keeps the learner aware that the screen is a bounded view of a larger mathematical object.

3. Missing roots outside the window

Consider f(x)=(x-2)(x-40). On a narrow window such as -5≤x≤10, the graph clearly shows the root at 2 but not the root at 40. A learner who trusts the image without the equation may report one real root.

The algebra immediately exposes both factors. The graph then becomes a way to inspect how those roots appear visually, not the authority that determines how many exist.

When the equation is not factorised, widen the window systematically or use theoretical information about the function. For a quadratic, algebra tells us there can be at most two real roots. For a polynomial, degree places a bound on the number of real roots. For periodic functions, one window can never represent every root unless a domain has already been restricted.

4. A root can hide between sampled points

Digital graphs are drawn from numerical evaluations. A plotting system samples values and joins or renders them according to an algorithm. If a function changes very rapidly over a tiny interval, a coarse sample can fail to display the behaviour faithfully.

School-level graphing tools are generally excellent for ordinary curriculum functions, but the learner should still understand the principle: the screen is generated numerically. A missing visual crossing is not a theorem that no crossing exists.

5. A repeated root may touch rather than cross

For f(x)=(x-3)², the graph touches the x-axis at x=3 and turns upward. A student trained to identify roots only by “where the graph crosses the axis” may miss the root entirely.

The correct definition is that a root is an input where f(x)=0. Crossing is one visual behaviour; touching is another. Algebraic multiplicity explains the difference.

For (x-3)²(x+1), the graph touches at 3 and crosses at -1. A graph can show this pattern beautifully once the learner knows what to look for.

6. A vertical scale can make distinct functions look identical

Plot f(x)=x² and g(x)=x²+0.01. On a vertical scale from -100 to 100, the two curves may appear to lie on top of each other. Zoom near the x-axis and the difference becomes visible.

The graph did not change. The display resolution changed what the eye could distinguish. Visual coincidence on a particular scale is therefore not proof that two functions are equal.

7. A horizontal scale can hide oscillation

Periodic functions can oscillate many times in a wide interval. If too much horizontal range is compressed into a small screen, individual cycles may merge visually or appear as a dense band. The learner may underestimate the number of roots, turning points or sign changes.

When studying periodic behaviour, choose a window related to the period. For trigonometric functions, know the period algebraically first, then use the graph to visualise repeated structure.

8. Asymptotes can be mistaken for crossings

For f(x)=1/(x-2), the function is undefined at x=2. Depending on the plotting engine, the two branches may be joined visually across a vertical asymptote or may appear extremely close to the line x=2. The graph must not persuade the learner that the function takes a value at the excluded input.

The domain decides whether a point belongs to the function. The display does not override the domain.

9. A hole can disappear on the screen

Consider

f(x)=(x²-1)/(x-1) with x≠1.

Algebra simplifies the expression to x+1 for every allowed input. The graph is therefore the straight line y=x+1 with a hole at (1,2). On a small screen, that missing point may be invisible.

A learner who sees an apparently continuous line might conclude the function is defined at x=1. The original formula and domain show otherwise. This is an ideal example of why graphical appearance cannot replace symbolic conditions.

10. Near-intersections can be visually misleading

Suppose two functions differ only slightly over the visible range. Their curves may seem to intersect because line thickness and pixel resolution overlap. Conversely, a genuine intersection may be difficult to see when the curves meet at a shallow angle.

Use the graph to locate an approximate region, then solve or verify f(x)=g(x) by an exact or controlled numerical method. The intersection tool on a calculator can provide a numerical candidate; substitution or residual checking then confirms whether the candidate fits the equations to the claimed accuracy.

11. Rounding can create a false zero

If a calculator displays a function value rounded to 0.000, the exact value may not be zero. It may be a small non-zero number hidden by the display precision.

For example, if the true value is 0.0004, a three-decimal display may show 0.000. Treating that as an exact root confuses numerical approximation with algebraic equality.

Record approximate roots with an approximation sign or an appropriate number of decimal places. Use exact forms when the problem requires exactness.

12. Numerical root tools return candidates within a method

A calculator’s solve or root command usually relies on numerical algorithms. The result can depend on a starting value, interval or nearby behaviour. A returned root is not automatically the complete solution set.

For a function with several roots, a solver may locate one nearby root and stop. The learner still needs mathematical reasoning to know how many roots may exist in the relevant domain and whether additional searches are required.

13. Example: one root found, another missed

Take f(x)=x³-4x. Factoring gives

x(x²-4)=x(x-2)(x+2),

so the exact roots are -2,0,2.

If a numerical root tool is started near x=1.8, it may return the root near 2. That answer is correct but incomplete. The solver has answered “find a nearby zero,” not “prove that these are all real zeros.”

14. Example: graph first, algebra second

Consider x²-5x+4=0. A graph of y=x²-5x+4 suggests roots near 1 and 4. That is useful evidence. Factorisation gives

(x-1)(x-4)=0,

which confirms the exact roots x=1,4.

Here the graph supports discovery and checking. The factorisation establishes exactness and completeness for the quadratic.

15. Example: graph suggests no root, discriminant explains why

For y=x²+2x+5, a standard window may show a parabola entirely above the x-axis. The discriminant is

b²-4ac=4-20=-16<0.

Therefore the quadratic has no real roots. The graph is consistent with the algebra; the discriminant supplies the general reason.

16. Example: a narrow window creates a false conclusion about monotonicity

A function can look increasing on a small interval even if it later turns. Consider f(x)=x³-3x. On a carefully chosen local interval, a section of the curve may appear only increasing or only decreasing. Across a wider domain the function has two turning points.

A visual claim such as “the function is increasing” must specify the interval. Calculus or exact comparison can establish monotonicity where required.

17. Example: zooming can manufacture confidence

Suppose the graph of two functions appears to meet at x≈1.73. Repeated zooming may make the intersection look more precise, but visual zoom alone does not turn the approximation into an exact value. If the equation has an exact form such as √3, a decimal image cannot reveal the symbolic structure by itself.

Use the graph to estimate; use algebra to identify exact structure when available.

18. Discrete data should not automatically be joined by a continuous curve

If a table records the number of buses arriving each hour, the input may be discrete time blocks or whole-number counts. Joining points with a smooth curve can visually suggest values between observations that were never measured or do not make sense.

The representation must match the mathematical object. A scatter plot, bar chart, step graph or discrete set of points may be more faithful than a continuous curve.

19. Interpolation and extrapolation are different risks

A fitted graph can estimate between observed values; that is interpolation. Extending beyond the observed range is extrapolation. The same line or curve may look persuasive in both regions, but extrapolation depends more strongly on whether the model continues to hold.

The graph shows what the model predicts. It does not prove the real system follows the model outside the data range.

20. A graph of a model is not validation of the model

Suppose a linear model fits five points reasonably well. Plotting the line through them shows the fit visually. It does not establish that the chosen variables capture the real mechanism, that measurement errors are small, or that the model will predict future observations reliably.

Model validation requires comparison with evidence and assumptions. This belongs to BTT’s modelling routes, not merely graph drawing.

21. The window audit

Before drawing a conclusion from a graph, record:

22. The evidence ladder for graph-based claims

Level 1: visual plausibility. The curve appears to cross near a value, rise over an interval or approach an asymptote.

Level 2: numerical check. Evaluate the function near the suspected point, inspect a table or compute a residual.

Level 3: symbolic verification. Substitute the candidate exactly, factor, rearrange, use an identity or apply a relevant exact method.

Level 4: completeness argument. Explain why no required cases are missing. For a quadratic this may follow from factorisation or the quadratic formula; for a restricted trigonometric equation it may require periodicity and interval analysis.

Level 5: proof. For a universal statement, use a valid general argument rather than a finite set of plotted examples.

23. A graph can disprove a universal claim with one genuine counterexample

If someone claims f(x)>0 for all real x, a graph showing a point below the x-axis can suggest a counterexample. But verify the candidate numerically or symbolically before declaring the claim false. Once a genuine input with f(x)≤0 is established, one counterexample is enough to refute the universal claim.

This asymmetry matters: many plotted examples cannot prove a universal statement, but one verified counterexample can disprove it.

24. Graphing transformations: use invariants

For g(x)=f(x)+3, every output increases by 3, so the graph shifts vertically upward by 3. For h(x)=f(x-2), the input needed to produce each old output increases by 2, so the graph shifts right by 2.

A learner who memorises arrows may confuse the direction. A learner who tracks input-output relationships can reconstruct the transformation. Use the graph as a check of the reasoning, not as a substitute for it.

25. Graphing reciprocal behaviour: domain first

For f(x)=1/x, the excluded input x=0 determines a vertical asymptote. For g(x)=1/(x-3)+2, the vertical asymptote shifts to x=3 and the horizontal asymptote to y=2.

If the learner enters a formula incorrectly, the screen can still produce a smooth and convincing graph. Predicting asymptotes before plotting gives an independent structural check.

26. Graphing exponential and logarithmic inverses

The graphs of y=2ˣ and y=log₂x are reflections across y=x because the functions are inverses on their appropriate domains. A graph can make this relationship visible.

The domain and range swap: the exponential has all real inputs and positive outputs; the logarithm has positive inputs and all real outputs. The visual reflection should agree with those symbolic facts.

27. A practical root-search protocol

1. State the equation and domain.
2. Rearrange to f(x)=0 if useful.
3. Use algebra to identify structure, possible factorisation, symmetry or root bounds.
4. Choose a window based on that information rather than accepting the default blindly.
5. Locate candidate roots graphically or numerically.
6. Refine approximations only to the accuracy required.
7. Substitute or evaluate residuals.
8. Explain why the root list is complete over the required domain.

28. A practical intersection protocol

To solve f(x)=g(x) graphically, plot both functions and identify intersections. Then:

• check that both functions are defined at each candidate;
• record approximate coordinates honestly;
• solve f(x)-g(x)=0 symbolically when exact answers are required;
• inspect whether other intersections may lie outside the window;
• state the domain over which completeness is claimed.

29. A practical turning-point protocol

For a quadratic, completing the square or using symmetry gives exact turning-point information. A graph can confirm the position. For a calculus problem, derivative conditions may establish stationary points and classify local behaviour. A graphical maximum/minimum tool can provide a numerical estimate, but the method required by the course still matters.

30. Numerical artefact casebook

Case A — apparent crossing. Two thick curves overlap visually near a point. Repair: evaluate their difference and solve for an actual zero.

Case B — clipped turning point. The y-window cuts off the top of a curve, making it look monotonic. Repair: expand the y-range or use symbolic turning-point information.

Case C — hidden hole. A removable discontinuity is smaller than a pixel. Repair: inspect the original formula and domain.

Case D — false zero from rounding. Display shows 0.000. Repair: increase precision or evaluate the exact expression.

Case E — missed oscillation. Wide x-range compresses many cycles. Repair: use period information and a narrower interval.

Case F — solver finds one root. Numerical command returns a nearby root. Repair: use structure and domain to search for all required roots.

31. Original worked problem: missing root

A learner plots f(x)=x²-12x+20 in a window from x=-5 to x=5. The graph shows a crossing near x=2 and the learner reports one root.

Factor:

x²-12x+20=(x-2)(x-10).

The roots are 2 and 10. The second root lies outside the window. The error is not a graphing error; it is an inference error. The learner turned “one root visible here” into “one root exists.”

32. Original worked problem: touching root

Graph g(x)=x²-8x+16=(x-4)². The curve touches the axis at 4 and turns upward. If a root is defined as a crossing, the learner misses it. If a root is defined correctly by g(x)=0, x=4 is immediately recognised.

33. Original worked problem: undefined point

Consider h(x)=(x²-4)/(x-2). The screen may show what looks like the line y=x+2. But the original function is undefined at 2. The correct graph is the line with a hole at (2,4).

The equation h(x)=4 has no solution at x=2 because h(2) is undefined, even though the simplified line would pass through that coordinate.

34. Original worked problem: approximate intersection

Suppose f(x)=x² and g(x)=2ˣ. A graph can reveal approximate intersections and guide numerical work. It does not by itself tell the learner whether every intersection has been found or whether a displayed decimal is exact.

The appropriate next move depends on the course: numerical solving, interval reasoning, calculus or a stated approximation method. The graph chooses a region; the mathematics justifies the conclusion.

35. Original worked problem: data and model

A dataset contains points (1,3),(2,5),(3,7),(4,9). A line y=2x+1 fits them exactly. Plotting the line and points confirms this data relationship. It does not prove that every future observation follows the same rule unless the context supplies a reason that the relationship is structurally exact.

A pattern in four observations is evidence for a model, not proof of a universal law.

36. When graphing is the right primary method

Graphing may be the intended method when a problem asks for an approximate solution from a curve, an interpretation of real data, a visual comparison of functions or a numerical estimate from a graph. The phrase “a graph is not proof” should not be turned into “graphs are inferior.”

Graphs are legitimate mathematical objects and can be central to rigorous reasoning. The key is matching the evidence to the claim.

37. When symbolic work should lead

Symbolic work is especially useful when the task asks for exact roots, an identity, a proof, a complete solution set, a domain condition or a reason that must hold for all inputs. In these cases, a graph can support intuition and checking but should not replace the required argument.

38. When numerical work should lead

Some equations do not have convenient elementary exact solutions. Numerical methods are then not second-rate substitutes; they are the appropriate tools. The learner must still specify the interval, accuracy, method and verification standard required by the problem.

39. Calculator discipline: predict before pressing

Before using a graphing or numerical tool, write one prediction:

• expected sign;
• rough root interval;
• likely number of turning points;
• known asymptote;
• expected symmetry;
• domain restriction.

If the screen contradicts the prediction, do not immediately discard the mathematics. Check the input, mode, window and assumptions. A wrong entry can produce a very convincing wrong graph.

40. Calculator discipline: preserve the entered expression

Write the mathematical expression before entering it. This creates an audit trail. If the graph is surprising, compare the device entry with the written expression. Many “calculator problems” are scope problems: a missing bracket, incorrect exponent, degree/radian mismatch or truncated denominator.

41. Degree and radian mode can radically change trigonometric graphs

A graph of sin x interpreted in degrees behaves on a very different horizontal scale from one interpreted in radians. If the course expects radians and the calculator is in degree mode, roots and periods may appear in unexpected places.

Mode is therefore part of the mathematical setup, not a device detail to ignore.

42. Logarithmic scale and transformed axes

Some graphs use non-linear scales. A straight line on a log-transformed axis does not mean the original variables have a simple linear relationship. Always read the axis definition before interpreting slope or distance.

43. Statistical graphs carry their own evidential limits

A scatter plot can suggest association, clusters, outliers and possible non-linearity. It cannot by itself establish causation. A fitted regression line summarises a pattern under a model. It does not turn every point into a prediction rule with certainty.

This is the same principle at a different level: visible structure is evidence whose strength depends on the claim.

44. Graphs in inequalities

To solve f(x)>g(x) graphically, look for regions where the graph of f lies above the graph of g. Intersections define candidate boundaries. But excluded points, open endpoints and hidden intersections outside the window still matter.

Translate the visual region back into interval notation and test boundary conditions explicitly.

45. Graphs in optimisation

A plotted model can suggest where a maximum or minimum occurs. If the domain is discrete—such as whole numbers of items—the visually highest point of a continuous curve may not itself be an admissible choice. Check nearby allowed values.

Optimisation is therefore a good place to connect graph behaviour with domain restrictions.

46. Graphs in probability and simulation

A simulated frequency graph can show convergence or variability across trials. It does not prove a theoretical probability. The simulation provides empirical evidence under a programmed model. The probability calculation comes from the sample space and assumptions.

Label simulations clearly and do not present a synthetic run as measured learner data or real-world research.

47. Graphs in sequences

A sequence is defined on discrete integer indices. Plotting points can reveal trends, but joining them continuously may suggest values between terms that the sequence never takes. A recurrence or explicit formula establishes the actual terms.

48. Graphs and inverse functions

A function needs an appropriate one-to-one domain before its inverse is also a function. The horizontal-line test can provide visual evidence. But the domain restriction should be stated mathematically.

For f(x)=x², restricting to x≥0 allows the inverse f⁻¹(x)=√x. Without the restriction, the original function is not one-to-one over all real numbers.

49. Graphs and piecewise functions

Piecewise graphs are especially vulnerable to endpoint mistakes. Open and closed circles represent excluded and included boundary values. A screen with small markers may obscure the distinction, so always read the piecewise conditions.

50. The “what can the screen not establish?” question

After every graph-based conclusion, ask what remains unproved. Examples:

• I see two roots here, but I have not shown there are no others outside the window.
• The curves look equal, but I have not proved the functions are identical.
• The values appear to approach a limit, but the graph alone does not establish the limit.
• The scatter plot shows association, but it does not establish causation.
• The fitted line predicts a value, but I have not validated the model beyond the data.
• The root is displayed as 1.414, but I have not shown the exact value is √2.

This single question turns passive viewing into mathematical judgement.

51. Planted graph errors

Error 1: “There is no root because the graph does not cross the axis.” The window is -10≤x≤10, but algebra gives a root at 25. Repair: distinguish the displayed interval from the whole domain.

Error 2: “The function is continuous because the graph is an unbroken line.” The original rational expression has a removable hole. Repair: inspect the domain.

Error 3: “The solution is exactly 1.73 because the intersection tool says 1.73.” Repair: label the value approximate unless exact reasoning establishes otherwise.

Error 4: “The function has one root because the solver returned one.” Repair: identify the solver’s local job and investigate completeness.

Error 5: “The data prove the trend continues.” Repair: separate fitted behaviour within the data range from extrapolation.

52. Fresh task 1: window control

Graph f(x)=(x+1)(x-12) first on -5≤x≤5, then on -5≤x≤15. Before changing the window, predict what new feature should appear. Explain why the first view was incomplete rather than wrong.

53. Fresh task 2: exactness

Graph y=x²-2 and estimate the positive root. Then solve algebraically. Compare the displayed decimal with the exact value √2. State which answer form is appropriate if the question requests an exact value and which is appropriate if it asks for three decimal places.

54. Fresh task 3: hole and domain

Graph y=(x²-9)/(x-3). Explain why the image may resemble y=x+3. State the excluded input and describe the missing point. Then answer whether x=3 can solve (x²-9)/(x-3)=6.

55. Fresh task 4: repeated root

Compare f(x)=(x-2)² with g(x)=(x-2)(x+2). Explain the difference between touching and crossing at roots. Connect the graph behaviour to factor multiplicity.

56. Fresh task 5: solver completeness

For h(x)=x³-5x, use a graph to estimate all real roots, then factor exactly. Explain why one successful numerical root search would not be enough to establish the complete set.

57. Fresh task 6: inequality

Use the graphs of y=x² and y=2x+3 to estimate where x²≤2x+3. Then solve algebraically and express the exact interval. Explain what role the graph played and what role the algebra played.

58. Fresh task 7: modelling

Use the synthetic points (1,2),(2,4.1),(3,5.9),(4,8.2). Plot them and consider a linear model. Describe what the graph supports, what it does not prove, and what additional evidence would be needed before using the model far outside the observed range.

59. Fresh task 8: trigonometric window

Graph y=sin x over one period and over many periods. Solve sin x=1/2 in a specified interval. Explain why the specified interval is necessary for a complete finite answer.

60. Worked answers: what matters

For Task 1 the roots are -1 and 12; the narrow window can display only the first. For Task 2 the exact positive root is √2; a decimal is an approximation. For Task 3 the original domain excludes x=3; the line has a removable hole there, so 3 is not an allowed solution. For Task 4 the squared factor gives a repeated root at 2 and a touch-turn behaviour, whereas the two distinct linear factors give crossings at -2 and 2. For Task 5, x³-5x=x(x²-5), so roots are 0,±√5. For Task 6, rearrange to x²-2x-3≤0, factor to (x-3)(x+1)≤0, giving -1≤x≤3.

61. A student-facing graph audit card

1. What function or data did I actually enter?
2. What domain is relevant?
3. What x-window and y-window am I seeing?
4. What features do I predict before looking?
5. What features are visible?
6. Could any required feature be outside the window or hidden by scale?
7. Are displayed coordinates exact or approximate?
8. What independent check can I use?
9. Have I established completeness over the required domain?
10. What can the graph not prove?

62. Tutor feedback language

Replace “don’t trust the calculator” with precise feedback:

• The graph is useful, but your window does not cover the domain you are making a claim about.
• You found a candidate root; now verify it in the equation.
• You have shown a local pattern, not a global result.
• The curve appears continuous, but the original formula excludes this input.
• Your numerical answer needs an approximation sign and stated accuracy.
• The solver returned one root; explain why the solution set is complete.
• The graph suggests an intersection; use an independent method to confirm it.

63. When the graph and algebra disagree

Do not choose whichever result you prefer. Audit both routes. Check the algebra for sign, factorisation and domain errors. Check the graph entry, mode, window and scale. If necessary, evaluate the function at selected points.

Disagreement between representations is valuable because it reveals that at least one model, entry or inference needs repair.

64. Transfer to calculus

In calculus, graphs can suggest where derivatives are positive, negative or zero, and where concavity changes. The derivative then supplies a mathematical account of rates and stationary points. A plotted curve can guide the analysis; derivative reasoning can verify and refine it.

65. Transfer to statistics

In statistics, graphs reveal distribution shape, spread, skew, clusters and outliers. Numerical summaries then quantify aspects of that pattern. Neither representation alone should dominate automatically. A mean without a graph can hide structure; a graph without a summary can make comparison imprecise.

66. Transfer to examination work

When graphing technology is allowed, the examiner may still require working, reasoning or exact results. The device can support a route without replacing the mathematical explanation. Follow the specific examination rules and marking requirements for the relevant course.

67. Transfer without the tool

A strong learner should be able to predict important graph features before plotting and explain them after the screen is removed. Give a changed function and ask for intercepts, domain, end behaviour, transformations or approximate shape from structure alone. Then use the graph as feedback.

68. Release test

The graphing capability is transfer-ready when the learner can:

• choose a window for a stated purpose;
• identify when a root may be hidden;
• recognise touching roots and discontinuities;
• distinguish exact and approximate information;
• explain numerical solver limits;
• verify candidate intersections independently;
• preserve domain restrictions;
• separate model fit from model validity;
• state what a particular graph does and does not establish;
• repeat the reasoning on a fresh function without a topic label.

69. Continue through BTT

Use How to Use a Calculator in Maths for input, exactness, estimation and device control. Use How to Check Maths Answers for independent verification. Use Reversible Steps, Lost Information and Extraneous Solutions when a graphical candidate must be checked against domain and algebraic transformation logic.

A graph is powerful because it lets us see Mathematics. Mathematical judgement begins when we also know what the picture cannot see for us.