FRQ Model Center | Carey AP Precalculus Practice Lab

Official sample · 2027Graphing calculator required

Retention model · the updated threshold and domain task

2026–27 course · May 2027 preparation

College Board · Fall 2026 CED sample · Question p. 148; scoring pp. 155–156 (PDF pp. 155, 162–163)Source

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 $t=0$ . They were tested at the end of each of the following 12 months. At $t=3$ months, the score was 70.84.

The score is modeled by $R(t)=a+b\ln(t+1)$ , where $R(t)$ is measured in points and $t$ is the number of months since the initial test.

2027 update

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.

BUILD YOUR RESPONSE

Your responses

Work through each part. Reveal the model answer and check its scoring points when you are ready.

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A(i)

Part A(i)

1 point

Use the data to write two equations that can be used to determine $a$ and $b$ .

Your response Equations, evidence, and a conclusion

Learn this partMethod notes · A(i)

PART A(i) · WHAT TO DO

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.

Translate every observation into a model equation

FRQ2 · Non-periodic models

Translate every observation into a model equation

What is being tested

Represent observed input-output pairs in the exact supplied model form, including shifts and branch conditions.

1Recognize

Write equations that can be used to find the constants.

2Write

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.

Build your response

Because [first input-output observation] and [second observation], the equations are [full model with first pair substituted] and [full model with second pair substituted].

3Avoid

Using $ab^t$ when the model specifies $ab^{t+1}$ ; taking a logarithm of time instead of the shifted argument; selecting the wrong piecewise branch.

Teaching example

If $F(t)=ab^{t+2}$ , $F(0)=12$ , and $F(2)=48$ , write $ab^2=12$ and $ab^4=48$ . For $G(t)=a+b\ln(t+1)$ with $G(0)=9$ and $G(2)=6$ , write $a+b\ln1=9$ and $a+b\ln3=6$ .

Why the reasoning works

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.

Compare answer & self-assess

WORKED ANSWER

$a+b\ln(0+1)=75$ and $a+b\ln(3+1)=70.84$ .

Check each scoring point

Mark a point earned only when your own response contains the required evidence. An unreviewed point contributes zero to this display.

1 ptBoth equations correctly connect the data to the model.

Earns credit

$a+b\ln1=75$ , $a+b\ln4=70.84$ .

Where credit is lost

$a+b\ln0=75$ , $a+b\ln3=70.84$ .

My response: Both equations correctly connect the data to the model.UnreviewedEarnedNot yet

A(ii)

Part A(ii)

1 point

Find $a$ and $b$ as decimal approximations.

Your response Equations, evidence, and a conclusion

Learn this partMethod notes · A(ii)

PART A(ii) · WHAT TO DO

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, store, and check the fitted constants

FRQ2 · Non-periodic models

Solve, store, and check the fitted constants

What is being tested

Solve the model system while maintaining enough precision for later predictions.

1Recognize

Find the constants as decimal approximations.

2Write

For $y=ab^t$ , divide two positive-output equations to get $b^{t_2-t_1}=y_2/y_1$ , choose the positive base, and substitute back for $a$ . Apply any given time shift when recovering $a$ .
For $y=a+b\ln(t+s)$ , subtract the equations to isolate $b$ , then find $a$ . 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.

Build your response

[Eliminate/subtract/divide] to obtain [one-parameter equation]. Thus [constants], to the requested precision. Substitution reproduces [given observations].

3Avoid

Reporting only a regression formula when constants are requested, choosing a negative exponential base, or reusing rounded coefficients in a sensitive later computation.

Teaching example

For $ab=6$ and $ab^3=24$ , division gives $b^2=4$ . A positive exponential base gives $b=2$ , then $a=3$ . Check: $3\cdot2=6$ and $3\cdot2^3=24$ .

Why the reasoning works

Elimination uses the model structure instead of guessing. Stored precision prevents an early approximation from changing a later root or threshold.

Compare answer & self-assess

WORKED ANSWER

$a=75$ , $b=\dfrac{70.84-75}{\ln4}=-3.000805685\ldots\approx-3.001$ . Thus $R(t)\approx75-3.001\ln(t+1)$ . Keep the unrounded value of $b$ in subsequent calculations.

Check each scoring point

Mark a point earned only when your own response contains the required evidence. An unreviewed point contributes zero to this display.

1 ptCorrect values for both parameters a and $b$ .

Earns credit

$a=75,\ b\approx-3.001$ .

Where credit is lost

$a=75,\ b=+3.001$ .

My response: Correct values for both parameters a and

b

.UnreviewedEarnedNot yet

B(i)

Part B(i)

1 point

Use the observed data to find the average rate of change from $t=0$ to $t=3$ , in points per month. Give a decimal approximation and show the computation.

Your response Equations, evidence, and a conclusion

Learn this partMethod notes · B(i)

PART B(i) · WHAT TO DO

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.

Compute a secant slope and interpret its units

FRQ2 · Non-periodic models

Compute a secant slope and interpret its units

What is being tested

Distinguish total output change from average output change per input unit.

1Recognize

Average rate of change, followed by a computation or contextual interpretation.

2Write

Use the requested endpoints and write $m=(F(t_2)-F(t_1))/(t_2-t_1)$ . 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.

Build your response

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].

3Avoid

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.

Teaching example

A tank has $12$ liters at hour $0$ and $4$ liters at hour $4$ . Its average rate is $(4-12)/(4-0)=-2$ liters per hour. Over hours 0 through 4, the amount decreased by an average of 2 liters per hour.

Why the reasoning works

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.

Compare answer & self-assess

WORKED ANSWER

$\dfrac{R(3)-R(0)}{3-0}=\dfrac{70.84-75}{3}=-1.386666\ldots\approx-1.387$ points per month.

Check each scoring point

Mark a point earned only when your own response contains the required evidence. An unreviewed point contributes zero to this display.

1 ptCorrect average rate of change with the requested computation.

Earns credit

$(70.84-75)/(3-0)\approx-1.387$ points per month.

Where credit is lost

$70.84-75=-4.16$ points per month.

My response: Correct average rate of change with the requested computation.UnreviewedEarnedNot yet

B(ii)

Part B(ii)

1 point

Interpret the average rate of change from B(i) in the context of this study.

Your response Equations, evidence, and a conclusion

Learn this partMethod notes · B(ii)

PART B(ii) · WHAT TO DO

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.

Compute a secant slope and interpret its units

FRQ2 · Non-periodic models

Compute a secant slope and interpret its units

What is being tested

Distinguish total output change from average output change per input unit.

1Recognize

Average rate of change, followed by a computation or contextual interpretation.

2Write

Use the requested endpoints and write $m=(F(t_2)-F(t_1))/(t_2-t_1)$ . 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.

Build your response

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].

3Avoid

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.

Teaching example

A tank has $12$ liters at hour $0$ and $4$ liters at hour $4$ . Its average rate is $(4-12)/(4-0)=-2$ liters per hour. Over hours 0 through 4, the amount decreased by an average of 2 liters per hour.

Why the reasoning works

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.

Compare answer & self-assess

WORKED ANSWER

From month 0 to month 3, the group's score decreased, on average, by approximately 1.387 points per month.

Check each scoring point

Mark a point earned only when your own response contains the required evidence. An unreviewed point contributes zero to this display.

1 ptInterpret the negative average rate in context and units.

Earns credit

During the first 3 months, the group's score decreased on average by about 1.387 points per month.

Where credit is lost

The group loses exactly 1.387 points in every month.

My response: Interpret the negative average rate in context and units.UnreviewedEarnedNot yet

C(i)

Part C(i)

1 point

The study leaders want to use $R$ 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.

Your response Equations, evidence, and a conclusion

Learn this partMethod notes · C(i)

PART C(i) · WHAT TO DO

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.

Turn a stopping rule into a threshold and domain

FRQ2 · Non-periodic models

Turn a stopping rule into a threshold and domain

What is being tested

Solve for the relevant boundary input, then explain a contextual domain rather than giving only an algebraic domain.

1Recognize

A model is usable until a boundary, or current official Q2 C(i) and C(ii).

2Write

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.

Build your response

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.

3Avoid

Giving only $t>-1$ for a shifted logarithm, omitting the contextual stopping rule, confusing total and additional time, or assigning a finite zero to positive exponential decay.

Teaching example

If $F(t)=8(1/2)^t$ is usable while $F(t)\ge1$ for $t\ge0$ , solve $8(1/2)^T=1$ to get $T=3$ . The domain is $0\le t\le3$ . If the rule were $F(t)>1$ , the upper endpoint would be excluded.

Why the reasoning works

The equation finds where the rule changes status; the inequality and context determine which side and whether equality belongs in the domain.

Compare answer & self-assess

WORKED ANSWER

$R(t)=65$ , so $75-3.000805685\ldots\ln(t+1)=65$ . Therefore $t=\exp\!\left(\dfrac{-10}{-3.000805685\ldots}\right)-1=27.006548786\ldots\approx27.007$ months.

Check each scoring point

Mark a point earned only when your own response contains the required evidence. An unreviewed point contributes zero to this display.

1 ptAn equation for the 65-point threshold AND the correct time.

Earns credit

$75-3.000805686\ln(t+1)=65$ , giving $t\approx27.007$ months (27.006 is also shown in the official guide).

Where credit is lost

$t=28.007$ months.

My response: An equation for the 65-point threshold AND the correct time.UnreviewedEarnedNot yet

C(ii)

Part C(ii)

1 point

Explain how your time from C(i) gives an appropriate domain restriction for $R$ . Refer to the context.

Your response Equations, evidence, and a conclusion

Learn this partMethod notes · C(ii)

PART C(ii) · WHAT TO DO

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.

Turn a stopping rule into a threshold and domain

FRQ2 · Non-periodic models

Turn a stopping rule into a threshold and domain

What is being tested

Solve for the relevant boundary input, then explain a contextual domain rather than giving only an algebraic domain.

1Recognize

A model is usable until a boundary, or current official Q2 C(i) and C(ii).

2Write

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.

Build your response

3Avoid

Giving only $t>-1$ for a shifted logarithm, omitting the contextual stopping rule, confusing total and additional time, or assigning a finite zero to positive exponential decay.

Teaching example

If $F(t)=8(1/2)^t$ is usable while $F(t)\ge1$ for $t\ge0$ , solve $8(1/2)^T=1$ to get $T=3$ . The domain is $0\le t\le3$ . If the rule were $F(t)>1$ , the upper endpoint would be excluded.

Why the reasoning works

The equation finds where the rule changes status; the inequality and context determine which side and whether equality belongs in the domain.

Compare answer & self-assess

WORKED ANSWER

The initial test is at $t=0$ . The decreasing model reaches 65 points at about $t=27.007$ months and predicts less than 65 after that. Therefore an appropriate contextual domain is approximately $0\leq t\leq27.007$ months.

Check each scoring point

Mark a point earned only when your own response contains the required evidence. An unreviewed point contributes zero to this display.

1 ptExplain the contextual domain using the initial time and 65-point threshold.

Earns credit

Use $0\leq t\leq27.007$ months: the test begins at 0, and predictions fall below the allowed 65 points after this endpoint.

Where credit is lost

The domain is $t>-1$ because the logarithm must be defined.

My response: Explain the contextual domain using the initial time and 65-point threshold.UnreviewedEarnedNot yet

Practice workspace

Your responses

1Recognize

2Write

3Avoid

Teaching example

Why the reasoning works

Check each scoring point

Earns credit

Where credit is lost

1Recognize

2Write

3Avoid

Teaching example

Why the reasoning works

Check each scoring point

Earns credit

Where credit is lost

1Recognize

2Write

3Avoid

Teaching example

Why the reasoning works

Check each scoring point

Earns credit

Where credit is lost

1Recognize

2Write

3Avoid

Teaching example

Why the reasoning works

Check each scoring point

Earns credit

Where credit is lost

1Recognize

2Write

3Avoid

Teaching example

Why the reasoning works

Check each scoring point

Earns credit

Where credit is lost

1Recognize

2Write

3Avoid

Teaching example

Why the reasoning works

Check each scoring point

Earns credit

Where credit is lost