Mathematics

Section Overview

The Mathematics section of the digital SAT consists of 44 questions across 70 minutes, split into two adaptive modules of 22 questions each (35 minutes per module). Approximately 75% of questions are multiple-choice with four answer options; the remaining 25% are student-produced response (grid-in) questions where you enter a numerical answer.

A calculator is permitted throughout the entire Mathematics section. The Bluebook app includes a built-in Desmos graphing calculator.

Content Domains

Domain	Approximate Weight	Question Count
Algebra	~35%	13-15
Advanced Math	~35%	13-15
Problem Solving and Data Analysis	~15%	5-7
Geometry and Trigonometry	~15%	5-7

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Algebra

Linear Equations and Inequalities

One-Variable Linear Equations:

The standard form is $ax + b = 0$, with solution $x = -\frac{b}{a}$ (for $a \neq 0$).

Steps to solve:

1. Distribute and combine like terms.
2. Isolate the variable on one side.
3. Verify by substituting back into the original equation.

Systems of Linear Equations:

A system of two linear equations in two variables:

$$\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}$$

Three solution methods:

• Substitution — Solve one equation for one variable and substitute into the other. Best when one variable has a coefficient of 1.
• Elimination — Multiply equations to align coefficients, then add or subtract to eliminate a variable. Best when coefficients are readily matched.
• Graphical — The solution is the intersection point of the two lines. Useful for estimating or verifying.

Key concepts:

• A system has one solution if the lines intersect (different slopes).
• A system has no solution if the lines are parallel (same slope, different intercepts).
• A system has infinitely many solutions if the lines are identical (same slope and intercept).

Linear Functions and Graphs

The equation of a line: $y = mx + b$

• $m$ = slope $= \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$
• $b$ = y-intercept (value of $y$ when $x = 0$)

Forms of a linear equation:

Form	Equation	Best Used For
Slope-intercept	$y = mx + b$	Graphing, identifying slope/intercept
Point-slope	$y - y_1 = m(x - x_1)$	Writing equation given a point
Standard	$Ax + By = C$	Determining intercepts, systems

Parallel lines have equal slopes ($m_1 = m_2$). Perpendicular lines have slopes that are negative reciprocals ($m_1 \cdot m_2 = -1$).

Quadratic Functions

Standard form: $f(x) = ax^2 + bx + c$

Vertex form: $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex.

Factored form: $f(x) = a(x - r_1)(x - r_2)$, where $r_1, r_2$ are the x-intercepts (roots).

Converting between forms:

To convert from standard to vertex form, complete the square:

$$ax^2 + bx + c = a\left(x^2 + \frac{b}{a}x\right) + c = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c$$

The vertex is at $x = -\frac{b}{2a}$.

The discriminant $D = b^2 - 4ac$ determines the nature of the roots:

Condition	Roots	Graph
$D > 0$	Two distinct real roots	Parabola crosses x-axis twice
$D = 0$	One repeated real root	Parabola touches x-axis
$D < 0$	No real roots (two complex)	Parabola does not cross x-axis

Factoring quadratics: Look for integer factor pairs of $ac$ that sum to $b$.

The quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Use when factoring is difficult or when the roots are not integers.

Polynomial Operations

Addition and subtraction: Combine like terms.

Multiplication: Use the distributive property (FOIL for binomials).

$$(a + b)(c + d) = ac + ad + bc + bd$$

Special products:

$$(a + b)^2 = a^2 + 2ab + b^2$$ $$(a - b)^2 = a^2 - 2ab + b^2$$ $$(a + b)(a - b) = a^2 - b^2$$

Division: Use polynomial long division or synthetic division. The Remainder Theorem states that the remainder when $P(x)$ is divided by $(x - c)$ is $P(c)$. The Factor Theorem states that $(x - c)$ is a factor of $P(x)$ if and only if $P(c) = 0$.

Functions and Their Graphs

A function $f$ maps each input $x$ to exactly one output $f(x)$.

Domain: The set of all valid inputs. Range: The set of all possible outputs.

Key function families:

Function	Equation	Graph Shape	Key Feature
Linear	$f(x) = mx + b$	Straight line	Constant rate of change
Quadratic	$f(x) = ax^2 + bx + c$	Parabola	Vertex, axis of symmetry
Exponential	$f(x) = a \cdot b^x$	Exponential curve	Horizontal asymptote
Absolute value	$f(x) = \|x\|$	V-shape	Vertex at origin
Square root	$f(x) = \sqrt{x}$	Half parabola	Domain: $x \geq 0$

Transformations:

• $f(x) + k$ — vertical shift up by $k$
• $f(x) - k$ — vertical shift down by $k$
• $f(x + h)$ — horizontal shift left by $h$
• $f(x - h)$ — horizontal shift right by $h$
• $-f(x)$ — reflection over x-axis
• $f(-x)$ — reflection over y-axis
• $af(x)$ — vertical stretch by factor $a$ (if $|a| > 1$) or compression (if $0 < |a| < 1$)

Inequalities

Linear inequalities: Solve like equations, but reverse the inequality sign when multiplying or dividing by a negative number.

• $ax + b > 0 \implies x > -\frac{b}{a}$ (if $a > 0$)
• $ax + b > 0 \implies x < -\frac{b}{a}$ (if $a < 0$)

Absolute value inequalities:

• $|x| < c \implies -c < x < c$ (for $c > 0$)
• $|x| > c \implies x < -c$ or $x > c$ (for $c > 0$)

Quadratic inequalities:

1. Find the roots of the corresponding equation.
2. Test a value in each interval determined by the roots.
3. Select intervals where the inequality holds.

Exponential and Logarithmic Functions

Exponential growth: $f(x) = a(1 + r)^x$, where $r > 0$ is the growth rate. Exponential decay: $f(x) = a(1 - r)^x$, where $0 < r < 1$ is the decay rate.

Logarithms are the inverse of exponentials:

$$\log_b a = c \iff b^c = a$$

Properties of logarithms:

$$\log_b(xy) = \log_b x + \log_b y$$ $$\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y$$ $$\log_b(x^n) = n \log_b x$$ $$\log_b b = 1, \quad \log_b 1 = 0$$

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Advanced Math

Complex Numbers

A complex number is $z = a + bi$, where $a$ is the real part, $b$ is the imaginary part, and $i = \sqrt{-1}$.

Operations:

Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$

Multiplication: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

Division: Multiply numerator and denominator by the conjugate of the denominator:

$$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i$$

Modulus: $|z| = \sqrt{a^2 + b^2}$

Powers of $i$ cycle every 4: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$.

Trigonometry

Right triangle trigonometry:

$$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}$$

Key identities:

$$\sin^2\theta + \cos^2\theta = 1$$ $$\tan\theta = \frac{\sin\theta}{\cos\theta}$$ $$\sin(2\theta) = 2\sin\theta\cos\theta$$ $$\cos(2\theta) = \cos^2\theta - \sin^2\theta$$

Unit circle: A circle of radius 1 centred at the origin. Key values at standard angles:

Angle ($^\circ$)	$0$	$30$	$45$	$60$	$90$	$180$	$270$
$\sin\theta$	$0$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$	$0$	$-1$
$\cos\theta$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$	$-1$	$0$

Radian measure: $\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}$

Common radian values: $30^\circ = \frac{\pi}{6}$, $45^\circ = \frac{\pi}{4}$, $60^\circ = \frac{\pi}{3}$, $90^\circ = \frac{\pi}{2}$, $180^\circ = \pi$.

Circle Equations

Standard form: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the centre and $r$ is the radius.

Expanded form: $x^2 + y^2 + Dx + Ey + F = 0$

To convert from expanded to standard form, complete the square for both $x$ and $y$:

$$x^2 + Dx = \left(x + \frac{D}{2}\right)^2 - \frac{D^2}{4}$$ $$y^2 + Ey = \left(y + \frac{E}{2}\right)^2 - \frac{E^2}{4}$$

Arc length: $L = r\theta$ (where $\theta$ is in radians)

Arc area (sector): $A = \frac{1}{2}r^2\theta$ (where $\theta$ is in radians)

Polynomial Equations

The Fundamental Theorem of Algebra: Every polynomial of degree $n$ has exactly $n$ roots (counting multiplicity and complex roots).

Relationship between roots and coefficients:

For $ax^3 + bx^2 + cx + d = 0$ with roots $r_1, r_2, r_3$:

$$r_1 + r_2 + r_3 = -\frac{b}{a}, \quad r_1 r_2 + r_1 r_3 + r_2 r_3 = \frac{c}{a}, \quad r_1 r_2 r_3 = -\frac{d}{a}$$

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Problem Solving and Data Analysis

Ratios and Proportions

A ratio $a:b$ expresses a relationship between two quantities. A proportion states that two ratios are equal: $\frac{a}{b} = \frac{c}{d}$.

Strategy: Set up a proportion and cross-multiply: $ad = bc$.

Unit rates: Express quantities per single unit (e.g., miles per hour, cost per item).

Percentages

Basic calculations:

• Finding a percentage: $x\%$ of $n = \frac{x}{100} \cdot n$
• Percentage change: $\frac{\text{new} - \text{original}}{\text{original}} \times 100\%$
• Percentage increase: new = original $\times (1 + r)$
• Percentage decrease: new = original $\times (1 - r)$

Successive percentage changes do not directly add. For example, a 20% increase followed by a 20% decrease yields:

$$(1.20)(0.80) = 0.96 \implies 4\% \text{ decrease overall}$$

Descriptive Statistics

Measure	Definition
Mean	Sum of values divided by the count: $\bar{x} = \frac{\sum x_i}{n}$
Median	Middle value when data is sorted
Mode	Most frequently occurring value
Range	Maximum minus minimum
Standard deviation	Average distance of each value from the mean
IQR	Third quartile minus first quartile ($Q_3 - Q_1$)

Key relationships:

• For a symmetric distribution, mean $\approx$ median.
• For a right-skewed distribution, mean > median.
• For a left-skewed distribution, mean < median.

Effect of transformations:

• Adding a constant $c$ to every value: mean increases by $c$, standard deviation unchanged.
• Multiplying every value by $c$: mean multiplied by $c$, standard deviation multiplied by $|c|$.

Probability

Basic probability: $P(A) = \frac{\text{favourable outcomes}}{\text{total outcomes}}$

Complement: $P(A^c) = 1 - P(A)$

Union (or): $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Intersection (and): $P(A \cap B) = P(A) \cdot P(B)$ if $A$ and $B$ are independent.

Conditional probability: $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$

Two events are independent if and only if $P(A \mid B) = P(A)$.

Scatterplots and Modelling

Correlation coefficient $r$:

• $r = 1$: perfect positive linear relationship
• $r = -1$: perfect negative linear relationship
• $r = 0$: no linear relationship

Line of best fit (least squares regression): $\hat{y} = mx + b$

The coefficient of determination $r^2$ represents the proportion of variance in $y$ explained by the linear relationship with $x$.

Exponential modelling: When data grows or decays by a constant percentage, use $y = ab^x$.

To determine whether a linear or exponential model is more appropriate:

• If the first differences (consecutive differences in $y$) are roughly constant, use linear.
• If the ratios of consecutive $y$-values are roughly constant, use exponential.

Unit Conversions

Strategy: Use dimensional analysis (multiply by conversion factors equal to 1).

$$12 \text{ inches} \times \frac{1 \text{ foot}}{12 \text{ inches}} \times \frac{1 \text{ yard}}{3 \text{ feet}} = 1 \text{ yard}$$

Common conversions to know:

Quantity	Relationships
Length	1 foot = 12 inches; 1 yard = 3 feet; 1 mile = 5280 feet
Weight	1 pound = 16 ounces; 1 ton = 2000 pounds
Volume	1 cup = 8 fl oz; 1 pint = 2 cups; 1 quart = 2 pints; 1 gallon = 4 quarts
Metric	1 km = 1000 m; 1 m = 100 cm; 1 kg = 1000 g

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Calculator Strategies

Approved Calculators

Most graphing calculators are approved for the SAT, including:

• TI-84 Plus (all editions)
• TI-Nspire (non-CAS)
• Casio fx-CG50
• HP Prime

Not allowed: Calculators with CAS (Computer Algebra System) capabilities, QWERTY keyboards, or internet access. The TI-Nspire CAS is not permitted.

When to Use the Calculator

Use it for:

• Arithmetic with large or messy numbers (fractions, decimals, roots).
• Graphing functions to find intercepts, intersections, or maxima/minima.
• Checking solutions to equations.
• Statistical calculations (mean, standard deviation, regression).
• Converting between units.

Avoid it for:

• Simple arithmetic you can do mentally (mental math is faster and less error-prone).
• Problems where the algebra is straightforward (setting up the equation is the hard part).
• Problems where the calculator would not help (reading comprehension in word problems, identifying the correct formula).

Desmos Tips

The built-in Desmos calculator in Bluebook is powerful:

• Graph equations by typing y = mx + b and adjusting parameters with sliders.
• Find intersections by clicking the intersection point of two graphs.
• Regression — type a table of values, then use y1 ~ mx1 + b for linear or y1 ~ a*b^x1 for exponential regression.
• Absolute value — type abs(x) or use the keyboard shortcut.
• Square root — type sqrt(x) or use the radical button.