Review and reference

Separate mistakes in distribution from mistakes in structure. Rehearse sign control for the first; for the second, identify the common factor, square pattern, or coefficient relationship before expanding.

State the operation before applying a rule: multiplication, division, or a power of a power. Then state the domain condition. Check a radical simplification at a negative input whenever the problem allows one.

Name the target: roots, number of roots, vertex input, or extreme output. Choose factored form, the discriminant, or vertex form accordingly. If a parameter controls the leading coefficient, test when it becomes zero before calling the equation quadratic.

Write a restriction line above every rational or radical equation. Treat solutions of the transformed equation as candidates until they survive substitution into the original. An empty solution set is a legitimate result.

Write three labels beside the model: initial or anchor value, multiplier, and time interval. The exponent must count periods in the same units as the model. Distinguish the whole amount from the amount above a fixed baseline.

Use a known point as an input-output statement. Solve for the new input before adjusting the output. For restricted ranges, compare endpoints with any vertex inside the allowed interval; do not quote the unrestricted range automatically.

A candidate must satisfy every original relationship and every domain restriction. When subtracting or substituting changes the degree, classify the equation that actually remains instead of applying a memorized intersection rule.

Use this reference to retrieve a relationship, not to replace the conditions that justify it. The official Math reference sheet is geometry-focused; the quadratic formula and exponent laws below are not supplied there.^[3]

Algebraic structure

Exponents and radicals

For positive bases a, b and real exponents r, s, the following laws are valid:

a^ra^s = a^r+s, $\frac{{\text{a}}^{\text{r}}}{{\text{a}}^{\text{s}}}$ = a^r−s, (a^r)^s = a^rs, (ab)^r = a^rb^r, ($\frac{\text{a}}{\text{b}}$)^r = $\frac{{\text{a}}^{\text{r}}}{{\text{b}}^{\text{r}}}$, a^−r = $\frac{\text{1}}{{\text{a}}^{\text{r}}}$.

For a > 0 and positive integer n, a^m/n = $\sqrt[\text{n}]{{\text{a}}^{\text{m}}}$ = ($\sqrt[\text{n}]{\text{a}}$)^m. Also a⁰ = 1 for a ≠ 0. Integer-exponent rules extend to negative bases when all expressions are defined; arbitrary fractional-exponent rewrites need additional care.

$\sqrt{{\text{u}}^{\text{2}}}$ = |u|, $\sqrt[\text{3}]{{\text{u}}^{\text{3}}}$ = u, $\sqrt{\text{ab}}$ = $\sqrt{\text{a}}$$\sqrt{\text{b}}$ for a, b ≥ 0.

Three false shortcuts In general, (u+v)² ≠ u²+v², $\sqrt{\text{u + v}}$ ≠ $\sqrt{\text{u}}$+$\sqrt{\text{v}}$, and a^r+s ≠ a^r +a^s. Distribution applies to multiplication over addition, not to every operation written outside parentheses.

For f(x) = ax² + bx + c with a ≠ 0:

Restricted domains change extrema. On a closed interval, compare both endpoints with the vertex if its input belongs to the interval. For integer inputs near a continuous vertex, compare the neighboring allowed integers and respect any endpoint constraints.

The parameter checkpoint For A(k)x² + B(k)x + C(k) = 0, handle A(k) = 0 first. If the remaining linear coefficient is nonzero, there is one solution. If both variable coefficients are zero, the constant equation determines whether there are no solutions or infinitely many. Use a discriminant only in the A(k) ≠ 0 case.

Exponential models

For f(t) = Ab^t with A > 0, b > 0:

f(0) = A, $\frac{\text{f(t + h)}}{\text{f(t)}}$ = b^h, b = ($\frac{\text{f(}{\text{t}}_{\text{2}}\text{)}}{\text{f(}{\text{t}}_{\text{1}}\text{)}}$)^{1/(t₂−t₁)} (t₂ ≠ t₁).

Growth by p% uses b = 1 + p/100; decay by p% uses b = 1 − p/100, with 0 < p < 100 for a positive decay factor. In Aq^t/d, q is the factor per d time units, and q^1/d is the factor per one unit (d > 0). In Aq^(t−t₀)/d, the anchor value is A at t = t₀. For f(t) = c + Ab^t, the ratio rule applies to f(t) − c, not to f(t) itself. Successive percentage-change factors multiply; the corresponding percentages do not generally add.

A decision table for unfamiliar questions

A calculator is a tool for choosing, checking, and accelerating a method. It should not silently choose the mathematics for you. Bluebook’s embedded calculator offers Desmos graphing and scientific options; calculators are permitted throughout Math. Follow the current official device policy, including the prohibition on CAS functionality.^[5]

1. Decide what the graph should answer

A root: graph y = f(x) and find where it meets the x-axis.

An equation f(x) = g(x): graph both y = f(x) and y = g(x) and inspect intersections. Alternatively, graph y = f(x) − g(x) and find its zeros. Use the common original domain.

An extreme value: identify the vertex or compare values at relevant allowed inputs. A continuous vertex is not automatically a feasible answer to an integer model.

A requested coordinate: read the correct coordinate and then evaluate any requested expression, such as x + y.

2. Enter expressions with deliberate grouping

Use parentheses around compound numerators, denominators, and exponents. The entries (x+1)/(x-2) and x+1/x-2 represent different expressions. Similarly, 2^(x+1) differs from 2^x+1. To evaluate a negative input in a square, enter the input in parentheses.

For a worked system such as 7.07, enter the two original rules separately. The intersection (4, 25) requires a window that reaches a relatively large positive y-value. A standard window could display only the other intersection. The algebraic degree and an intentional expanded window provide checks on completeness.

3. Use tables for exact input choices

A table is especially useful when the question restricts the input to integers. In Desmos, add a table and use the input column to list the values you want to test. A function expression such as f(x₁) can populate the output column from those inputs.^[6]

For the integer revenue model in 3.14, compare the allowed integer inputs on either side of the continuous vertex. For the threshold in 5.15, evaluate the last input below the threshold and the first input above it. A rounded approximate crossing is not the same as a proof that a particular integer is the first to satisfy a strict inequality.

A good division of labor

Use algebra to establish the domain, degree, or exact parameter condition. Use a graph to locate or interpret the result. Use a table to test discrete inputs. Use substitution to verify the final answer in the original relationships.

A missing point may be invisible. In 4.02, the canceled factor removes an input. A plotted curve can look continuous around the hole, especially at an ordinary scale. Read the original denominator, not just the picture.

Near tangency can look like tangency. A graph may make two close intersections appear to be one. For an exact parameter question such as 7.12, derive the discriminant condition and then check it visually. A slider provides conjectures, not an exhaustive proof.

A decimal is not the symbolic expression. A display close to 1.4142 does not by itself establish $\sqrt{\text{2}}$. Keep exact radicals and fractions during algebra, and use approximations only when appropriate to the target.

The wrong transformation can still draw a plausible curve. Check a mapped point using the inside-input equation. In 6.07, the correct input follows from 2x − 6 = 4; guessing the shift direction is less reliable.

A valid algebraic root can fail a context. A graph cannot decide whether the input must represent a nonnegative time, a positive dimension, or a whole-number count unless those constraints are explicitly supplied.

5. Enter the answer the question actually requests

For student-produced response questions, follow the on-screen directions. An exact fraction is useful when it fits. Do not enter a mixed number; use an improper fraction or a decimal. Omit units, dollar signs, percent signs, and commas. The published directions allow up to five characters for a positive answer, or six including the minus sign for a negative answer. When a decimal does not fit, follow the stated truncation or rounding instructions rather than assuming two decimal places suffice.^[3]

For example, 2/3 is exact and fits; 0.67 is an unnecessarily coarse approximation. When a question requests the greater root, enter that root only. When it requests a percentage, enter the percentage value, not its multiplier: a 12% increase corresponds to an entry of 12 when the question asks for the percent, not 0.12 or 1.12.

Final five‐second check

Is this the requested variable or expression? Is the value in the original domain? Does the context permit it? Is the entry exact or sufficiently precise? Have I omitted extra notation the response field does not accept?

Practice environment. Use the Desmos College Board graphing calculator practice environment to become comfortable with the testing interface. Recheck official calculator guidance before your test date.^[5]

The example numbers below are clickable. Mixed-question numbers link to the matching Guide question in mixed practice (for example, 8 opens M08). Every section also has a five-question exit check. A blank mixed-question entry means the skill is practiced in the lessons and exit checks rather than assigned its own item in the 42-question mixed set.

Sections 1 and 2: rewriting tools

A focused repair session

Choose one row, attempt its examples with worked solutions closed, and say why each operation is valid. Then attempt the linked mixed question without looking at the row label. A method that works only after the topic is named is not yet fully transferable.

An important distinction. “No real solution” can arise in different ways: a negative quadratic discriminant, a nonnegative quantity equated to a negative one, a contradiction after valid operations, or the rejection of every transformed candidate by the original domain. Name which mechanism applies.

Turn an error into a transferable rule

Domain error. You accepted a canceled denominator zero or an extraneous radical candidate. Repair: write restrictions before algebra and check the original relationship.

Structure error. You expanded when roots, symmetry, or a repeated expression could expose the target. Repair: identify the best form before computing.

Equivalence error. You divided by a possibly zero quantity, lost a square-root branch, or used a power rule outside its valid domain. Repair: name the condition that makes each transformation reversible.

Modeling error. You confused a rate with a multiplier, one period with several, or a baseline with the whole amount. Repair: annotate input units, output units, and the anchor point.

Target or execution error. You found the right intermediate quantity but entered the wrong one, miscopied a sign, or trusted a limited window. Repair: write the requested quantity at the top and verify it at the end.

A repeatable mastery cycle

First pass: learn the concept and attempt the worked question before opening its worked solution. Mark the first point at which you needed help.

Second pass: re-solve selected questions from a blank page after a gap. Explain the domain, method choice, and check aloud or in writing. Recognition from a remembered answer is not the same as reconstruction.

Transfer pass: attempt exit checks and mixed questions without the section labels as hints. Sort errors by cause. Revisit an example that teaches the missing decision, then try a different question.

Readiness check: you can solve unfamiliar questions accurately, justify restrictions, select efficient methods, and identify when a calculator display is incomplete. Use these as learning criteria, not a promised scaled score.

Three sentences worth being able to finish

“I chose this representation because...” “This operation is valid provided that...” “The answer satisfies the original problem because...”

For each entry, write the earliest mistake, not just the final wrong answer. Keep the corrected principle short enough to apply to a new problem.

Entry 1 Question: __________ Date: __________

First incorrect step or missing decision:

Corrected principle and its condition:

Example to revisit: __________ New problem: __________ Recheck: __________

Entry 2 Question: __________ Date: __________

First incorrect step or missing decision:

Corrected principle and its condition:

Example to revisit: __________ New problem: __________ Recheck: __________

Entry 3 Question: __________ Date: __________

First incorrect step or missing decision:

Corrected principle and its condition:

Example to revisit: __________ New problem: __________ Recheck: __________

Entry 4 Question: __________ Date: __________

First incorrect step or missing decision:

Corrected principle and its condition:

Example to revisit: __________ New problem: __________ Recheck: __________

One-line session summary: The decision I can now make more reliably is ________________________

Scope and policy check: September 10, 2026. The sources below support the testing context, official domain alignment, calculator guidance, and answer-entry discussion. Lessons, questions, mathematical derivations, and diagrams in this guide were created for this volume. Links are clickable; policies and interfaces can change.

[1] College Board. Advanced Math. Official broad skill categories and named nonlinear function families.

[2] College Board. Math Specifications. Domain descriptions and approximate question distribution.

[3] College Board. Assessment Framework for the Digital SAT Suite, version 3.01, August 2024. Supports the Advanced Math testing points, Math directions and reference sheet.

[4] College Board. How the SAT Is Structured. Standard section lengths, module structure, and question formats.

[5] College Board. SAT Suite of Assessments Calculator Policy. Permitted use, embedded calculator options, and device restrictions. Consult the current version before test day.

[6] Desmos. Tables. Official documentation for input-output tables and function-generated columns; page updated August 21, 2026.

How the guide is organized

The seven sections are instructional subdivisions, not a claim that College Board uses seven official Advanced Math subdomains. Algebraic structure appears repeatedly because the same relationship supports factoring, functions, and systems. Selected enrichment details, such as a factor/remainder shortcut or a graph’s removable hole, are taught as ways to understand polynomial and rational structure, not as guarantees of a particular question’s frequency.

How to interpret the practice

The book contains exactly 105 labeled worked examples, 35 section exit questions, and 42 mixed questions. Difficulty labels are editorial. The independent sets are not calibrated, equated, or adaptive and provide no official score conversion. A small focused set cannot test every skill variant; the lessons, worked examples, exit checks, and mixed sets should be used together. Independent publication. This resource is not affiliated with or endorsed by College Board or Desmos. SAT is a registered trademark of College Board. No paid or free commercial question bank was reproduced to create these original problems.