A synthetic calibration table gives outputs at inputs 0, 2, 4 and 6. You want a formula that fits the observed values and predicts another output. Begin by asking how much the output changes over each equal input step, rather than guessing a curve from its appearance.
Input x
Output y
Next output minus this output
0
5
13−5=8
2
13
45−13=32
4
45
101−45=56
6
101
No next observation
The input step is h=2. The first differences are 8,32,56. Their differences are 32−8=24 and 56−32=24. These are the second differences. A missing next observation is not a zero difference.
Predict
Predict a possible next first difference and output if the second difference continues to be 24. Then separate two claims: “a quadratic fits these four observations” and “the unknown process must be quadratic everywhere.” Which claim could the table alone justify?
Also predict whether the coefficient of x² equals 24/2=12. That familiar shortcut assumes an input step of 1. Here each step is 2; ignoring it changes the proposed formula.
Experiment
Continuing the observed pattern gives next first difference 56+24=80 and next output 101+80=181 at x=8. Now reconstruct the candidate rather than merely extending its table.
Write q(x)=ax²+bx+c. For equal steps h, the second difference is 2ah², as derived below. Here 24=2a·2², so a=3. The observation q(0)=5 gives c=5. From q(2)=13, obtain 3·4+2b+5=13, hence 2b=−4 and b=−2. Thus q(x)=3x²−2x+5. Check the remaining observations: q(4)=48−8+5=45 and q(6)=108−12+5=101. Finally q(8)=192−16+5=181 agrees with the continued-difference prediction.
The next graph adds a competing formula. Predict whether its dashed curve must follow the solid one after the last observed input.
Solid: q(x)=3x²−2x+5. Dashed: R(x)=q(x)−½x(x−2)(x−4)(x−6). Dots mark the four supplied observations. These are formula-generated curves, not extra measured data. Axis scales differ and are labeled; both models use the same scale.
Input
Quadratic q
Competing R
Status
0
5
5
Supplied observation
2
13
13
Supplied observation
4
45
45
Supplied observation
6
101
101
Supplied observation
8
181
−11
Different model predictions; not observed
At x=8 the product 8·6·4·2 is 384, so R(8)=181−384/2=−11. At the unmeasured input x=1, within the observed input range, q(1)=6 while R(1)=6−½(1)(−1)(−3)(−5)=13.5. A new measurement there can distinguish these candidates without extrapolating beyond the range. At every supplied input, one factor of the product is zero. Both formulas fit all observations exactly. The second model is a mathematical counterexample to uniqueness, not a recommendation that negative outputs are physically plausible.
Understand
Here is the equal-step calculation using only algebra. For q(x)=ax²+bx+c,
q(x+h)−q(x)=2ahx+ah²+bh.
At the next input x+h, the first difference becomes 2ah(x+h)+ah²+bh. Subtract the preceding first difference: the x, constant and b terms cancel, leaving 2ah². Consequently doubling the step multiplies a quadratic’s second difference by 4. The output differences are not average rates: divide each first difference by h to get the average rate over that interval. In the worked table those rates are 4,16,28 output units per input unit.
For a polynomial of degree d≥1 with nonzero leading coefficient and fixed nonzero step, taking a difference reduces its degree by one. A constant polynomial instead has every first difference zero; the zero polynomial has no degree under the usual convention. This follows by expanding the highest-power term: the xᵈ terms cancel, leaving a nonzero multiple of xᵈ⁻¹. Repeating d times gives a constant nonzero difference; the next differences are zero. Thus constant first differences suggest a linear candidate, constant nonzero second differences a quadratic candidate, and constant nonzero third differences a cubic candidate. Enough equally spaced observations must exist to compare differences at the claimed level. With only one computed second difference, there is no second one to compare it with.
The converse needs a restriction on the candidate family. Three distinct inputs uniquely determine a polynomial of degree at most 2 when a fitting quadratic exists: the difference of two candidates would be a polynomial of degree at most 2 with three distinct roots, forcing it to be zero. Four measurements can then check that chosen quadratic. But no finite table proves that the unrestricted underlying function belongs to that family.
For any distinct observed inputs x₁,…,xₙ, the product (x−x₁)⋯(x−xₙ) vanishes at every observed input. Adding any constant multiple of this product to a fitted polynomial preserves every observed output. Different constants generally change predictions elsewhere. More observations can challenge a candidate; any finite set still permits such alternative formulas if no additional assumptions constrain the model.
Apply
First diagnose unequal spacing. For x=0,1,3,4 and y=x², outputs are 0,1,9,16. Raw first differences are 1,8,7 and second differences are 7,−1. They are not constant, although the generating formula is quadratic. The equal-step theorem does not apply because the step lengths are 1,2,1. Do not declare the formula nonquadratic from those raw differences. Use a correctly specified model with the actual input coordinates or collect an equally spaced design; neither permission means silently moving the measured inputs.
Now reconstruct a fresh candidate from this table:
x
0
0.5
1
1.5
y
2
2.5
4
6.5
Find the second differences, recover a,b,c for a quadratic, and predict y at x=2. State what a new measurement at 2 could and could not establish.
Check the candidate after working
First differences are 0.5,1.5,2.5, giving second differences 1,1. Here h=0.5, so 1=2a(0.5)² and a=2. The input 0 gives c=2; input 0.5 then gives 2.5=0.5+0.5b+2, so b=0. The candidate is 2x²+2 and predicts 10 at x=2. A different exact value would refute this candidate as an exact generating formula. Agreement would support this additional check but would not prove global equality. Real measurements may also have noise or rounding, requiring a stated tolerance rather than an exact-match rule.
Continue
The solid quadratic has positive leading coefficient and grows without bound at either end of the full real line. The dashed polynomial has leading term −½x⁴, so it eventually decreases without bound at both ends. These incompatible end behaviors coexist with the same finite observations. End behavior follows from an assumed formula, not merely from a few increasing table values.
Record the candidate family, input spacing and domain of use with any prediction. If the calibration process is defined only for 0≤x≤6, neither an x=8 prediction nor an infinite-input claim describes an available experiment without extending the model’s domain. Choose a new admissible input, predict before measuring, and retain the result even if it contradicts the candidate. A fit, a successful new check and a justified model are different levels of evidence.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
At x=0,1,2,3, measured outputs are 3,6,13,24. Construct first and second differences, find a quadratic matching all four points, and state what the table does not prove.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Use f(x)=x² at x=0,1,3,6. Compute the first and second raw output differences, then calculate each secant rate. Explain why nonconstant raw second differences do not contradict the stipulated quadratic.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let q(x)=2x²+x+3. Construct a different polynomial agreeing with q at 0,1,2,3 but having value q(1/2)+9/16 at x=1/2. State its degree and evaluate the sign of the modification at 1/2.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A nonzero polynomial P is known in advance to have degree at most 3. Five equally spaced inputs have constant nonzero second differences. Explain why P must actually have degree 2. Then explain why dropping the degree bound makes that conclusion invalid from the same five samples.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Two candidate models are q(x)=x²+1 and r(x)=x²+1−x(x−1)(x−2)(x−3). Verify their agreement at 0,1,2,3. State both end limits for each unrestricted model and explain why finite matching cannot choose between those end behaviors.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A synthetic measurement table at x=0,1,2,3 is 1.0,2.1,5.0,10.2, with each output accurate only to ±0.2. A learner rejects every quadratic because the displayed second differences are not exactly equal. Evaluate the claim by checking the candidate x²+1 and discussing precision.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
