Review and reference

Worksheets and ledgers are optional reference records. Write on paper or keep your own notes; these records do not save progress on this website.

On this page

The practice question numbers below are the stable Guide question numbers shown on each question, even when the bank shuffles their order.

Quick reference: input and interpretation

Need	Build or inspect	Final check
Grouped fraction	$(a+b)/(c+d)$	The whole denominator remains below the fraction bar.
Negative substitution	$(-3)^2$	The square applies to the negative number, not just 3.
Fractional exponent	$9^(3/2)$	Both 3 and 2 are in the exponent.
Radical of a sum	$sqrt(a+b)$	All intended terms are under the radical.
Function evaluation	Define f(x), then evaluate f (k).	k is an input, not the output to solve for.
Angle from a ratio	Actual inverse-trig function	Degrees or radians? Inverse function or reciprocal?
Long calculation	Use exact fractions, π, and full factors.	Round only at the requested final step.
New problem	Inspect old definitions and table names.	A hidden curve can still have an active definition.

Graph feature decoder

The question asks for...	Report...
A solution of $f(x) = 0$	An allowed x-intercept input.
A solution of $f(x) = k$	An intersection input for $y = f(x)$ and $y = k.$
The solution of a two-variable system	The relevant ordered pair, or the requested expression of its coordinates.
A maximum or minimum value	The output, including domain restrictions.
When a maximum occurs	The input at the maximum.
The number of solutions	A justified count, not just visible dots in one window.

Four checks before a graph becomes an answer

Window: Is the relevant region visible? Domain: Is each candidate allowed? Precision: Is an approximate coordinate being mistaken for exactness? Target: Does the coordinate or derived value answer the wording?

Quick reference: tables, models, and responses

Task	Recipe	Meaning
Function table	Inputs $x 1;$ outputs $f(x 1)$	Evaluate the same rule repeatedly.
Linear regression	$y_1~m*x_1+b$	m is output units per input unit.
Quadratic regression	$y_1~ax_1^2+bx_1+c$	Fit only a justified model with enough independent information.
Exponential regression	$y_1~a*b^(x_1)$	b multiplies the output per one input unit.
Three-unit period	$y_1~a*b^(x_1/3)$	b is now the multiplier per three input units.
Residual	observed − predicted	Same input; difference in output units.
Weighted mean	$\frac{∑(value)(count)}{∑ count}$	Counts determine how often each value contributes.
Spread comparison	Translate or scale distances from the mean.	Adding a constant does not change spread.

Before trusting fitted parameters

Preserve data pairs. Use the intended input units. Remove conflicting definitions. Distinguish an exact model from a noisy fit. Reuse fitted parameters rather than shortened displays. A numerical fit does not prove causation or justify extrapolation by itself.

Before typing a numeric response

Use a fraction or decimal that fits the field. Enter only one answer, even when multiple answers are possible. Omit units, commas, dollar signs, percent signs, mixed-number notation, and symbolic radicals. Follow the displayed directions and any rounding specified in the question.^[13][14]

Exact value	Example entry	Common wrong turn
$7/12$	$7/12$ or $.5833$	Arbitrarily reducing to .58.
$-11/6$	$-11/6$ or $-1.833$	Forgetting the sign or typing a mixed number.
$\sqrt{7}$	$2.646$	Typing a root expression into a numeric field.
37.5% when asked for p in p%	$37.5$	Entering the multiplier .375.

Troubleshooting decision sheet

Symptom	Likely issue to inspect	Recovery move
An impossible negative length	Angle mode, leading minus, or wrong model	Check a known trig value; reconstruct the defining equation.
A plausible but unexpected decimal	Missing grouping or wrong target	Read numerator, denominator, and requested quantity aloud.
No graph appears	Entry error, disabled expression, or window scale	Inspect the expression; estimate landmarks; set both axis ranges.
Only one root seems visible	Off-screen root, tangency, or insufficient resolution	Use algebra to bound or count roots, then refine the view.
A point lies where the expression is undefined	Cancelled denominator or another restriction	Check the original domain; reject forbidden candidates.
Regression seems to ignore some data	Wrong columns, missing pairs, or fixed parameters	Audit rows, column names, and earlier definitions.
A mean differs from the expected value	Frequencies were ignored	Reconstruct total value and total count separately.
The answer is close but fails a check	Premature rounding, a scale issue, or wrong equation	Reuse exact expressions; substitute into the original problem.
The correct result was never recorded	Calculator work mistaken for a response	Enter it in the question and inspect answer status.
The testing app stops responding	Technical issue, not necessarily a math error	Raise your hand; follow official troubleshooting directions.^[20]

Do not fix a symptom by inventing a number

Taking an absolute value, moving a decimal point, or rounding to a convenient integer is not a repair unless the mathematics requires that operation. Locate the defect in the model, entry, interpretation, or response first.

Skills-to-examples map: operating the tools

Use this map after a mistake. Rework the linked example without looking at the solution, then attempt a fresh problem that changes the numbers or asks for a different quantity.

Skill to repair	Worked examples
Fraction grouping and nested templates	Example 1.01, Example 1.03, Example 1.15
Negative signs and substituted values	Example 1.02, Example 1.04
Powers, roots, and scientific notation	Example 1.05, Example 1.06, Example 1.09
Percent factors and precise geometry	Example 1.07, Example 1.08, Example 4.08
Degrees, radians, and inverse trig	Example 1.10, Example 1.11, Example 5.09
Function definitions and old parameters	Example 1.12, Example 1.14, Example 3.13
A complete grouped difference	Example 1.13
Roots, outputs, and requested coordinates	Example 2.01, Example 2.02, Example 2.03, Example 2.05
Vertices and tangent contacts	Example 2.04, Example 2.12, Example 5.05
Absolute values and circle intersections	Example 2.06, Example 2.07
Useful graph windows	Example 2.08, Example 2.09
Domains and extraneous candidates	Example 2.10, Example 2.11, Example 4.02, Example 4.03
Inequality regions and boundaries	Example 2.13
Nonlinear intersection counts	Example 2.14, Example 4.13
A parameter as the plotted unknown	Example 2.15

What mastery looks like

You can reproduce the method, explain the displayed output, identify a realistic failure mode, and complete a different problem without being told which feature or command to use.

Skills-to-examples map: models and decisions

Skill to repair	Worked examples
Function tables and candidate testing	Example 3.01, Example 3.02, Example 3.03
Linear rates and fitted lines	Example 3.04, Example 3.05, Example 3.06
Quadratic and exponential models	Example 3.07, Example 3.08, Example 3.09
Mean, median, weights, and spread	Example 3.10, Example 3.11, Example 3.12
Data sufficiency and input units	Example 3.14, Example 3.15
Original-equation checking	Example 4.01, Example 4.02, Example 4.03
Fractions, decimals, and response limits	Example 4.04, Example 4.05, Example 4.06, Example 4.07
Rounding and exact intermediate values	Example 4.08, Example 4.09
Integer constraints and percent targets	Example 4.10, Example 4.11
Requested expressions and small residuals	Example 4.12, Example 4.15
Identity versus numerical agreement	Example 4.14, Example 5.04, Example 5.11
Efficient mental or algebraic work	Example 5.01, Example 5.02, Example 5.03, Example 5.05
Parameter cases and exact root counts	Example 5.06
Appropriate numerical computation	Example 5.07, Example 5.08, Example 5.10
Study-design reasoning over button use	Example 5.12
Module review and technical recovery	Example 5.13, Example 5.14
The full model-to-response chain	Example 5.15

Connect this volume to the content guides

Use Algebra for linear structure and systems; Advanced Math for domains, transformations, and parameters; Geometry and Trigonometry for angle and measurement relationships; and Problem-Solving and Data Analysis for model interpretation, distributions, probability, and study design. Return to Essential Foundations when notation or arithmetic is the source of the tool error.

Digital rehearsal: from isolated skills to a full test

These are suggested practice activities, not scored diagnostic tests. Rehearse in the SAT-specific testing calculator and in Bluebook, not only on the unrestricted public Desmos website.^[3][4]

Rehearsal 1: interface fluency

Use an untimed Bluebook preview. Locate the calculator, reference sheet, question menu, timer, and mark-for-review control. Practice moving the calculator so it does not cover the relevant question text. Identify how to select an answer, cross out an option, and return to a marked question. Preview questions do not provide scores or answer feedback.^[2][16]

Then practice a small toolkit: a grouped fraction, a negative substitution, an inverse-trig calculation in the correct unit, a function evaluation, a quadratic’s roots, and a two-equation intersection. Explain each output before moving on. This is about control, not speed.

Rehearsal 2: deliberate method comparison

Choose six unfamiliar problems from the earlier content volumes. Before solving each, write “mental,” “algebra,” “graph,” or “table.” After completing it, try another valid approach when practical. Record setup time, interpretation mistakes, and whether the answer was exact or approximate.

Keep a method because it is reliable for you on new problems, not because a demonstration made it look effortless. Do not compare a first attempt using one method with a memorized second attempt using another and call the timing difference proof.

Rehearsal 3: integrated full-length practice

Take an official Bluebook full-length practice test under the conditions relevant to you, including approved timing settings where available. Review the resulting score and item feedback through the official practice workflow.^[16] During the test, use your scratch-paper target line, preserve calculator state intentionally, and transfer every intended response into its field.

Afterward, classify every miss and every unusually slow success. Was the obstacle conceptual knowledge, mathematical setup, calculator entry, graph interpretation, numerical precision, or navigation? Retest the specific behavior rather than merely rereading the explanation.

A useful success criterion

You can solve an unfamiliar problem, explain why the method fits, verify the original conditions, and navigate away knowing that the intended response was recorded. A correct number sitting only in the calculator is unfinished work.

Test-day readiness: device, policy, and workflow

This checklist separates durable habits from details that may change. Recheck the linked official pages before your administration. SAT Weekend and in-school testing may use different logistical instructions; follow your admission information and school or test-center guidance.

When	What to confirm
Well before testing	Confirm that your device type and operating system meet current Bluebook requirements. Supported categories include eligible Windows and Mac devices, iPads, and school-managed Chromebooks; do not assume a personal Chromebook or phone is eligible. Arrange school support or device lending early.^[18][19]
When exam setup opens	For SAT Weekend, complete the Bluebook exam-setup task in the days before the test and obtain the admission ticket. Verify the actual test date, location, arrival instructions, and your applicable settings.^[17]
Before packing	Charge the testing device fully and bring its permitted charging equipment. Check identification and admission requirements. A permitted handheld calculator is optional; inspect its current eligibility and remove prohibited stored material.^[17][1]
On arrival	Follow the proctor’s setup directions, use the provided scratch paper, and keep prohibited devices put away as directed. Do not treat a personal phone calculator as a backup.^[17][20]
During each module	Keep work organized by question. Use calculator entries as working space, not as recorded answers. Review blanks before leaving the module.
At test completion	Follow the app’s submission instructions and the proctor’s directions. Report a submission problem before leaving rather than assuming everything succeeded.^[17]

A separate check for a handheld calculator

Current policy prohibits CAS functionality; older advice or permission on another exam is not enough. Check the current feature rules rather than relying on a remembered model list. A familiar permitted calculator can be useful, but an unfamiliar backup adds its own risk.^[1]

Preparation is not permission to change testing rules

Practice notes, this guide, external browser calculators, stored documents, and online help are not resources to consult during the actual test. Follow the allowed-tool policy and the proctor’s instructions. A technical problem should go through the official support process.

Personal readiness check and study route

Check these skills without using the worked solutions

Check	Observable behavior	Repair route
□	I can enter a nested expression and explain its grouping.	Section 1
□	I can find and change angle mode, and distinguish inverse trig from a reciprocal.	Section 1
□	I can find roots, intersections, and vertices and name the requested coordinate.	Section 2
□	I can recover from an unhelpful graph window without guessing.	Section 2
□	I can create a table, fit a justified model, and retain paired data and units.	Section 3
□	I can distinguish a prediction, residual, mean, and frequency-weighted mean.	Section 3
□	I can reject excluded or extraneous values and keep precision through intermediate steps.	Section 4
□	I can enter fractions and decimals and check that the response field is populated.	Section 4
□	I can identify problems where a short manual method is more useful than graphing.	Section 5
□	I can use review tools and explain what to do if the app fails.	Section 5

Choose a route from evidence

New to the tools: work through the lessons in order and reproduce all examples. Complete each exit check before attempting the mixed sets.

Comfortable but inconsistent: start with the mixed sets. Use the map to repair the precise skills behind mistakes, then revisit the full digital workflow.

Accurate but slow: compare methods on unfamiliar problems. Focus on setup overhead, repeated entry, excessive zooming, and solving for quantities the question never requested.

Mastery is not a single raw score

A correct answer obtained by luck, a memorized output, or a rule you cannot explain is not secure mastery. Revisit a skill until you can transfer it to a changed problem with a sound check.

Printable error log and retest plan

Record one specific error per entry. A useful repair describes an action you can perform on the next problem.

Error entry 1

Date and question: __________

Requested quantity and my original method: __________
__________

Where the reasoning or tool use first went wrong: __________
__________

Category: concept / setup / entry / display / precision / response / timing

Replacement habit, stated as an action: __________
__________

Example to revisit: __________

New problem and retest date: __________

Evidence the repair worked: __________

Error entry 2

Date and question: __________

Requested quantity and my original method: __________
__________

Where the reasoning or tool use first went wrong: __________
__________

Category: concept / setup / entry / display / precision / response / timing

Replacement habit, stated as an action: __________
__________

Example to revisit: __________

New problem and retest date: __________

Evidence the repair worked: __________

Replace vague advice with a visible action

Instead of “be more careful,” write “record (x, y), underline the requested expression, and substitute before entering the response.” Instead of “learn Desmos,” write “practice setting both axis bounds from estimated landmarks.”

Official sources and version notes: 1–8

Official documentation was consulted in September 2026. Links are clickable; publisher wording and interface details may change. The mathematics, problems, figures, explanations, and suggested study routines in this guide are original. Source references support tool behavior, test structure, or policy, not a claim that these are official test questions.

[1] SAT Suite of Assessments Calculator Policy

College Board

Permitted calculator features, CAS prohibition, stored-material restrictions, and embedded calculators.

[2] Testing Tools

College Board / Bluebook

Calculator access, review tools, question navigation, zoom, and the timer display.

[3] Assessment Resources & FAQ

Desmos Studio

Assessment-specific configurations and the distinction between the scientific and graphing workspaces.

[4] Practice With Testing Calculators

Desmos Studio

How to select the relevant assessment calculator rather than assume public-site parity.

[5] Getting Started: Desmos Graphing Calculator

Desmos Studio

Graphing relationships, selecting points of interest, and basic expression-list use.

[6] Graph Settings

Desmos Studio

Axis bounds, scale controls, and degree/radian settings.

[7] Scientific Calculator

Desmos Studio

Numerical expression templates and the scientific-calculator interface.

[8] Tables

Desmos Studio

Data columns, computed columns, table entry, and plotted data.

Official sources: 9–16

[9] Regressions

Desmos Studio

Custom tilde regressions, fitted parameters, and model evaluation.

[10] Statistics

Desmos Studio

List statistics and the distinction between sample and population standard-deviation functions.

[11] Supported Functions

Desmos Studio

Reference for the function names used in mathematical recipes.

[12] Inequalities and Restrictions

Desmos Studio

Curly-brace restrictions, inequality shading, and strict versus non-strict boundaries.

[13] Student-Produced Responses

College Board

Numeric-response questions and the instruction to supply one answer.

[14] Official Math response-entry directions

College Board

Official test directions explain response-entry conventions. Follow the current Math directions in Bluebook for fractions, decimals, character limits, and omitted symbols.

[15] SAT Structure

College Board

Standard section timing, question counts, modules, and adaptive organization.

[16] Practice and Preparation

College Board / Bluebook

Untimed test previews, full-length practice, and the official practice-feedback workflow.

Official sources: 17–24

[17] SAT Weekend

College Board / Bluebook

Exam setup, admission materials, test-day device preparation, and completion procedures.

[18] Prepare Your Device

College Board / Bluebook

Device-readiness tasks and planning for school support or a loaned device.

[19] Approved Testing Devices

College Board / Bluebook

Current device categories and operating-system requirements; recheck before testing.

[20] What to Expect on Test Day

College Board

Module navigation limits, proctor assistance, prohibited-device rules, and submission guidance.

[21] How Are Scores Calculated?

College Board

Multistage adaptivity and the importance of doing your best across both modules.

[22] Functions

Desmos Studio

Function definitions, inputs, and repeated evaluations.

[23] Keyboard Shortcuts

Desmos Studio

Keyboard access to common editing operations; confirm controls on the actual testing device.

[24] Unresolved Detail in Plotted Equations

Desmos Studio

Limits of rendered implicit detail and why a plot is not an exact proof.

A source boundary that matters

The response-entry directions cited in [14] explain entry conventions. Rehearse the adaptive digital interface in Bluebook. Desmos help pages describe features; confirm assessment availability in the SAT-specific testing environment and Bluebook.