Recognize the four task families, build a clear response, and learn why an answer earns or loses credit.
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The current sample uses B to calculate and interpret an average rate of change. C now asks you to solve for a contextual threshold and use that threshold to explain a suitable domain restriction.
Older Drive questions may retain a linear estimate, a concavity justification, or a limit question. Those are useful practice, but their part labels do not define the revised official sample.
Official clarificationNon-periodic models: illustrated methods and guidance for each part.
Translate the conditions. Preserve the model. Explain the context.
Choose evidence with its input condition. Quadratic evidence requires a nonzero constant second difference; zero second differences are linear. A finite table can support a model choice; it does not prove how an arbitrary function behaves between the supplied inputs.
Calculator required. Current official sample: A = 2, B = 2, C = 2 points. It has no B(iii). Its C asks for a threshold and a contextual domain; historical Drive models keep their original A/B/C structure.
If a model must pass through given observations, substitute those conditions. Use regression only when fitting is requested or justified by the task.
For , write both observation equations before reporting and . Store full precision.
A calculator regression screen is not a substitute for requested equations.
Use the specified two times and the requested observed or modeled values.
. “From … to …, the score decreased on average by … points per month.”
Do not swap units, drop the direction, or call a secant rate an instantaneous rate.
Solve the threshold equation, then intersect mathematical and contextual restrictions. Include equality if the limit itself is permitted.
, . “The context begins at and the model is appropriate while , so .”
The algebraic logarithm domain is not the contextual answer.
An estimate, residual, range, or confidence claim is required only when that part asks for it.
Linear estimate: . Residual: observed − predicted. Range: corresponding outputs over the allowed domain.
Do not memorize the old B(iii) as a mandatory step of every new FRQ2.
| Recognize the task | Build the response | Evidence / boundary |
|---|---|---|
| Translate every observation into a model equation Write equations that can be used to find the constants. | Because [first input-output observation] and [second observation], the equations are [full model with first pair substituted] and [full model with second pair substituted]. | The setup shows which observations constrain the parameters. A time shift changes the exponent or logarithm argument even when it does not change elapsed time between observations. |
| Solve, store, and check the fitted constants Find the constants as decimal approximations. | [Eliminate/subtract/divide] to obtain [one-parameter equation]. Thus [constants], to the requested precision. Substitution reproduces [given observations]. | Elimination uses the model structure instead of guessing. Stored precision prevents an early approximation from changing a later root or threshold. |
| Compute a secant slope and interpret its units Average rate of change, followed by a computation or contextual interpretation. | The average rate is [shown difference quotient] [output units per input unit]. From [start] to [end], [quantity] [increased/decreased] by an average of [magnitude] [output units] per [input unit]. | The quotient is the slope of the line through the two endpoint observations. It compresses the net change over the interval into one constant rate. |
| Use the secant estimate the question requests Use the average rate from the previous part to estimate. | Using the secant through [endpoint pair], the estimate is [anchor output] + [rate] times ([target input] - [anchor input]) = [estimate], in [units]. | The average rate fixes the slope, and an endpoint fixes the line. A nonlinear curve can share those endpoints and still differ everywhere between them. |
| Explain secant comparisons and changing average rates A legacy B(iii) asks whether estimates are above/below the model, or whether later average rates are greater/less. | The estimates lie on the secant through [endpoints]. On [relevant span], [model] is [concave up/down], so [curve-versus-line or ordered-slope relationship]. Therefore [requested comparison]. | Concavity describes how slopes change. A chord averages those slopes between its endpoints, which also controls its position relative to the curve. |
| Turn a stopping rule into a threshold and domain A model is usable until a boundary, or current official Q2 C(i) and C(ii). | Solve [model equals boundary] to obtain [relevant T]. The model applies for [interval], because [start] and [why predictions after T violate the contextual rule]. Include or exclude equality according to the condition. | The equation finds where the rule changes status; the inequality and context determine which side and whether equality belongs in the domain. |
| Find the range on the contextual interval Possible output values during opening hours or a stated validity period. | On the contextual domain [input interval], [endpoint/vertex/branch checks] give [output range]. These are continuous model predictions in [units]; actual counts are whole numbers where appropriate. | An unrestricted range can include values at times that never occur in the scenario. Restricting the domain changes which extrema and outputs matter. |
| Explain confidence using the observed interval A question compares the reliability of predictions at two different times. | There is more confidence at [input] because it lies within the observed interval [bounds]. The other prediction extrapolates beyond the data and assumes [contextual pattern] continues. | A model is calibrated using observations in a particular region. Predictions outside it rely on additional untested persistence assumptions. |
| Define subtraction order before interpreting error Residuals, model-versus-estimate differences, or a maximum permitted prediction error. | Using [stated convention], the difference is [first value] - [second value] = [result]. Its sign means [which value is greater]; its magnitude is [amount] [units]. | Sign has meaning only after the subtraction order is fixed. Magnitude gives the size of the discrepancy; sign tells its direction. |
| Explain why an exponential becomes a semi-log line A legacy subpart asks about an exponential model on a semi-log graph. | Because the logarithm of the output equals [linear expression in time], log-output is linear in time. The semi-log graph is a line with [signed slope], consistent with [base condition]. | Logarithms convert multiplication into addition, turning a constant multiplicative change into a constant additive change in log-output. |
Change a value or trace a branch. Read the mathematical evidence produced by each step.
Name time and output units. Substitute each observation into the stated model, including any horizontal shift inside a logarithm.
Every equation contains the correct input and output.Solve parameters once, store the full model, and round only displayed final numerical answers.
Plug the observations back into the stored model.Compute the requested rate or estimate and write direction, quantity, interval, and units.
An answer such as −1.387 alone is not an interpretation.Solve the limiting condition and shade the permitted time interval. Explain why the model stops being appropriate.
A mathematical extension is not a real-world guarantee.. Move through time. The shaded interval satisfies the stated threshold.
67.303 points. Inside the stated useful domain.
Secant evidence: . Threshold evidence: , . Contextual conclusion: .
Retention model · the updated threshold and domain task · Graphing calculator required
College Board · Fall 2026 CED sample · Question p. 148; scoring pp. 155–156 (PDF pp. 155, 162–163)Students completed a class and then joined a year-long study of how much course content they retained. Their initial group test score was 75 out of 100 at . They were tested at the end of each of the following 12 months. At months, the score was 70.84.
The score is modeled by , where is measured in points and is the number of months since the initial test.
Use the data to write two equations that can be used to determine and .
and .
Both equations correctly connect the data to the model.
, .
, .
Find and as decimal approximations.
, . Thus . Keep the unrounded value of in subsequent calculations.
Correct values for both parameters a and b.
.
.
Use the observed data to find the average rate of change from to , in points per month. Give a decimal approximation and show the computation.
points per month.
Correct average rate of change with the requested computation.
points per month.
points per month.
Interpret the average rate of change from B(i) in the context of this study.
From month 0 to month 3, the group's score decreased, on average, by approximately 1.387 points per month.
Interpret the negative average rate in context and units.
During the first 3 months, the group's score decreased on average by about 1.387 points per month.
The group loses exactly 1.387 points in every month.
The study leaders want to use beyond 12 months. They judge the model appropriate until the predicted score is below 65. Write and solve an equation for the time when the predicted score equals 65. Give a decimal approximation in months.
, so . Therefore months.
An equation for the 65-point threshold AND the correct time.
, giving months (27.006 is also shown in the official guide).
months.
Explain how your time from C(i) gives an appropriate domain restriction for . Refer to the context.
The initial test is at . The decreasing model reaches 65 points at about months and predicts less than 65 after that. Therefore an appropriate contextual domain is approximately months.
Explain the contextual domain using the initial time and 65-point threshold.
Use months: the test begins at 0, and predictions fall below the allowed 65 points after this endpoint.
The domain is because the logarithm must be defined.
and .
Both equations correctly connect the data to the model.
, .
, .
Retention model · the updated threshold and domain task
Substitute the initial and later scores into the full shifted logarithmic expression; time zero is valid because the argument is t+1.
Use the initial observation to simplify one equation, then solve for the remaining coefficient. Store its full precision for the later threshold.
Show the score difference divided by the specified elapsed months, then report points per month with the requested precision.
Interpret the signed rate for the group score over the whole named interval; do not claim that every monthly score changed by exactly that amount.
Write the score model equal to the stated validity threshold, solve for the relevant time, and check with unrounded coefficients.
Use the threshold time from C(i) to restrict the model from its contextual start. Explain what the rule permits at equality and why later predictions fall outside it.
These are teaching frames, not a keyword checklist. Match the evidence requested by this part; memorized wording does not replace the required mathematics.
Represent observed input-output pairs in the exact supplied model form, including shifts and branch conditions.
Write equations that can be used to find the constants.
Identify the meaning, time origin, and units of both variables before substituting. If outputs are measured in thousands, keep that scale consistently.
Insert each observed input and its corresponding output into the full stated formula. Keep shifts inside exponents and logarithm arguments.
Use enough independent observations for the unknown constants. A quadratic usually needs three independent conditions; common two-parameter forms need two.
For piecewise models, choose the branch containing that input. If continuity is explicitly required, equate the branches at the junction; never assume continuity when it is not stated.
Present actual equations with equals signs. Do not replace a request for the setup with only the fitted parameter values.
Because [first input-output observation] and [second observation], the equations are [full model with first pair substituted] and [full model with second pair substituted].
Using when the model specifies ; taking a logarithm of time instead of the shifted argument; selecting the wrong piecewise branch.
If , , and , write and . For with and , write and .
The setup shows which observations constrain the parameters. A time shift changes the exponent or logarithm argument even when it does not change elapsed time between observations.
Three released examinations provide 12 more FRQs, three in each family. Use their year-specific scoring guides. The current CED sample above reflects the revised May 2027 format.
These are additional reference PDFs, separate from the integrated current samples, Drive models and Carey adaptations. The public archive currently lists student responses for 2024 and 2025; it lists questions and scoring guidelines for 2026. Official archive
2026–27 course · May 2027 preparation
Students completed a class and then joined a year-long study of how much course content they retained. Their initial group test score was 75 out of 100 at . They were tested at the end of each of the following 12 months. At months, the score was 70.84.
The score is modeled by , where is measured in points and is the number of months since the initial test.
In the revised official sample, B(iii) is removed. B now asks for average rate of change and its contextual interpretation. C asks for a threshold equation and a justified contextual domain. The sample still totals 6 points: A 2, B 2, C 2.
Older Drive models may retain linear-estimation, concavity, or limiting-value tasks. They remain useful historical practice; they are not the current sample's exact subpart sequence. The official family description permits more than logarithmic models, so do not assume every future Q2 will copy this context.
Work through each part. Reveal the model answer and check its scoring points when you are ready.
Saved in this browser · This browser · Self-assessment is not an AP score prediction.
Use the data to write two equations that can be used to determine and .
Substitute the initial and later scores into the full shifted logarithmic expression; time zero is valid because the argument is t+1.
These notes explain the method. Your response stays above; the selected question’s worked answer remains in its separate review.
Represent observed input-output pairs in the exact supplied model form, including shifts and branch conditions.
Write equations that can be used to find the constants.
Identify the meaning, time origin, and units of both variables before substituting. If outputs are measured in thousands, keep that scale consistently.
Insert each observed input and its corresponding output into the full stated formula. Keep shifts inside exponents and logarithm arguments.
Use enough independent observations for the unknown constants. A quadratic usually needs three independent conditions; common two-parameter forms need two.
For piecewise models, choose the branch containing that input. If continuity is explicitly required, equate the branches at the junction; never assume continuity when it is not stated.
Present actual equations with equals signs. Do not replace a request for the setup with only the fitted parameter values.
Because [first input-output observation] and [second observation], the equations are [full model with first pair substituted] and [full model with second pair substituted].
Using when the model specifies ; taking a logarithm of time instead of the shifted argument; selecting the wrong piecewise branch.
If , , and , write and . For with and , write and .
The setup shows which observations constrain the parameters. A time shift changes the exponent or logarithm argument even when it does not change elapsed time between observations.
and .
Mark a point earned only when your own response contains the required evidence. An unreviewed point contributes zero to this display.
, .
, .
Find and as decimal approximations.
Use the initial observation to simplify one equation, then solve for the remaining coefficient. Store its full precision for the later threshold.
These notes explain the method. Your response stays above; the selected question’s worked answer remains in its separate review.
Solve the model system while maintaining enough precision for later predictions.
Find the constants as decimal approximations.
For , divide two positive-output equations to get , choose the positive base, and substitute back for . Apply any given time shift when recovering .
For , subtract the equations to isolate , then find . For a quadratic, solve the linear system in its coefficients.
Store full-precision parameter values or exact expressions. Show requested rounded values, but use stored values in subsequent work.
Substitute the constants into every supplied observation. A mismatch signals a branch, time, scale, or rounding error.
Distinguish a parameter from an initial value: a shift can mean the coefficient named a is not the output at time zero.
[Eliminate/subtract/divide] to obtain [one-parameter equation]. Thus [constants], to the requested precision. Substitution reproduces [given observations].
Reporting only a regression formula when constants are requested, choosing a negative exponential base, or reusing rounded coefficients in a sensitive later computation.
For and , division gives . A positive exponential base gives , then . Check: and .
Elimination uses the model structure instead of guessing. Stored precision prevents an early approximation from changing a later root or threshold.
, . Thus . Keep the unrounded value of in subsequent calculations.
Mark a point earned only when your own response contains the required evidence. An unreviewed point contributes zero to this display.
.
.
Use the observed data to find the average rate of change from to , in points per month. Give a decimal approximation and show the computation.
Show the score difference divided by the specified elapsed months, then report points per month with the requested precision.
These notes explain the method. Your response stays above; the selected question’s worked answer remains in its separate review.
Distinguish total output change from average output change per input unit.
Average rate of change, followed by a computation or contextual interpretation.
Use the requested endpoints and write . Keep the same endpoint order in numerator and denominator.
Use observed endpoint data when the question says to use given data. Show the numerical quotient before reporting the result.
Attach output units per input unit, retaining scales such as thousands. A change in a percentage level is measured in percentage points, not relative percent change.
To interpret, identify the quantity, the entire time interval, the direction of change, and the average amount per unit time.
A negative result means decreasing on average across that interval. It does not assert a constant daily change or establish monotonicity at every instant.
The average rate is [shown difference quotient] [output units per input unit]. From [start] to [end], [quantity] [increased/decreased] by an average of [magnitude] [output units] per [input unit].
Giving only the output difference, saying the rate decreases when the quantity decreases, omitting interval or units, or claiming the exact same change each day.
A tank has liters at hour and liters at hour . Its average rate is liters per hour. Over hours 0 through 4, the amount decreased by an average of 2 liters per hour.
The quotient is the slope of the line through the two endpoint observations. It compresses the net change over the interval into one constant rate.
points per month.
Mark a point earned only when your own response contains the required evidence. An unreviewed point contributes zero to this display.
points per month.
points per month.
Interpret the average rate of change from B(i) in the context of this study.
Interpret the signed rate for the group score over the whole named interval; do not claim that every monthly score changed by exactly that amount.
These notes explain the method. Your response stays above; the selected question’s worked answer remains in its separate review.
Distinguish total output change from average output change per input unit.
Average rate of change, followed by a computation or contextual interpretation.
Use the requested endpoints and write . Keep the same endpoint order in numerator and denominator.
Use observed endpoint data when the question says to use given data. Show the numerical quotient before reporting the result.
Attach output units per input unit, retaining scales such as thousands. A change in a percentage level is measured in percentage points, not relative percent change.
To interpret, identify the quantity, the entire time interval, the direction of change, and the average amount per unit time.
A negative result means decreasing on average across that interval. It does not assert a constant daily change or establish monotonicity at every instant.
The average rate is [shown difference quotient] [output units per input unit]. From [start] to [end], [quantity] [increased/decreased] by an average of [magnitude] [output units] per [input unit].
Giving only the output difference, saying the rate decreases when the quantity decreases, omitting interval or units, or claiming the exact same change each day.
A tank has liters at hour and liters at hour . Its average rate is liters per hour. Over hours 0 through 4, the amount decreased by an average of 2 liters per hour.
The quotient is the slope of the line through the two endpoint observations. It compresses the net change over the interval into one constant rate.
From month 0 to month 3, the group's score decreased, on average, by approximately 1.387 points per month.
Mark a point earned only when your own response contains the required evidence. An unreviewed point contributes zero to this display.
During the first 3 months, the group's score decreased on average by about 1.387 points per month.
The group loses exactly 1.387 points in every month.
The study leaders want to use beyond 12 months. They judge the model appropriate until the predicted score is below 65. Write and solve an equation for the time when the predicted score equals 65. Give a decimal approximation in months.
Write the score model equal to the stated validity threshold, solve for the relevant time, and check with unrounded coefficients.
These notes explain the method. Your response stays above; the selected question’s worked answer remains in its separate review.
Solve for the relevant boundary input, then explain a contextual domain rather than giving only an algebraic domain.
A model is usable until a boundary, or current official Q2 C(i) and C(ii).
Translate the stopping rule into an equation at equality, such as a score threshold, zero quantity, or allowed difference between two predictions.
Solve using stored coefficients. Check candidate crossings in the original model, and select the first relevant boundary after the stated start when that is the rule.
Combine the contextual starting time, formula restrictions, and ending condition. State input units and distinguish additional elapsed time from time since the original origin.
Include the boundary when equality is allowed; exclude it for a strict positive/above condition. Continuous time should not be rounded down merely because observations were recorded at integer times.
A positive unshifted exponential decay never reaches exactly zero at finite time. Without a positive practical cutoff or another model, do not invent a finite stopping time.
The revised official sample separates the threshold equation in C(i) from the contextual explanation in C(ii); historical part C tasks can ask different things.
Solve [model equals boundary] to obtain [relevant T]. The model applies for [interval], because [start] and [why predictions after T violate the contextual rule]. Include or exclude equality according to the condition.
Giving only for a shifted logarithm, omitting the contextual stopping rule, confusing total and additional time, or assigning a finite zero to positive exponential decay.
If is usable while for , solve to get . The domain is . If the rule were , the upper endpoint would be excluded.
The equation finds where the rule changes status; the inequality and context determine which side and whether equality belongs in the domain.
, so . Therefore months.
Mark a point earned only when your own response contains the required evidence. An unreviewed point contributes zero to this display.
, giving months (27.006 is also shown in the official guide).
months.
Explain how your time from C(i) gives an appropriate domain restriction for . Refer to the context.
Use the threshold time from C(i) to restrict the model from its contextual start. Explain what the rule permits at equality and why later predictions fall outside it.
These notes explain the method. Your response stays above; the selected question’s worked answer remains in its separate review.
Solve for the relevant boundary input, then explain a contextual domain rather than giving only an algebraic domain.
A model is usable until a boundary, or current official Q2 C(i) and C(ii).
Translate the stopping rule into an equation at equality, such as a score threshold, zero quantity, or allowed difference between two predictions.
Solve using stored coefficients. Check candidate crossings in the original model, and select the first relevant boundary after the stated start when that is the rule.
Combine the contextual starting time, formula restrictions, and ending condition. State input units and distinguish additional elapsed time from time since the original origin.
Include the boundary when equality is allowed; exclude it for a strict positive/above condition. Continuous time should not be rounded down merely because observations were recorded at integer times.
A positive unshifted exponential decay never reaches exactly zero at finite time. Without a positive practical cutoff or another model, do not invent a finite stopping time.
The revised official sample separates the threshold equation in C(i) from the contextual explanation in C(ii); historical part C tasks can ask different things.
Solve [model equals boundary] to obtain [relevant T]. The model applies for [interval], because [start] and [why predictions after T violate the contextual rule]. Include or exclude equality according to the condition.
Giving only for a shifted logarithm, omitting the contextual stopping rule, confusing total and additional time, or assigning a finite zero to positive exponential decay.
If is usable while for , solve to get . The domain is . If the rule were , the upper endpoint would be excluded.
The equation finds where the rule changes status; the inequality and context determine which side and whether equality belongs in the domain.
The initial test is at . The decreasing model reaches 65 points at about months and predicts less than 65 after that. Therefore an appropriate contextual domain is approximately months.
Mark a point earned only when your own response contains the required evidence. An unreviewed point contributes zero to this display.
Use months: the test begins at 0, and predictions fall below the allowed 65 points after this endpoint.
The domain is because the logarithm must be defined.