SAT Mathematics Syllabus

Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics in SAT, not a summary of it.

1. 1

   Heart of Algebra

   4 topics

   • Linear Equations
   • Linear Inequalities
   • Linear Functions
   • Systems of Linear Equations
2. 2

   Problem Solving and Data Analysis

   6 topics

   • Ratios, Rates, and Proportions
   • Percentages
   • Units
   • Table Data
   • Graphs and Charts
   • Statistics and Probability
3. 3

   Passport to Advanced Math

   4 topics

   • Quadratic Equations
   • Functions
   • Polynomials
   • Exponential Growth and Decay
4. 4

   Additional Topics in Math

   3 topics

   • Geometry
   • Trigonometry
   • Complex Numbers

Mathematics flashcards for SAT

21 of 50 cards from the Mathematics deck — real questions with worked answers.

1. What is the slope-intercept form of a linear equation, and what does each variable represent?

   $y = mx + b$, where $m$ is the slope (rate of change) and $b$ is the $y$-intercept (the value of $y$ when $x = 0$).

2. How do you calculate the slope $m$ of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$?

   $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ the change in $y$ divided by the change in $x$.

3. What is the point-slope form of a linear equation?

   $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a known point on the line.

4. What is the standard form of a linear equation, and what is the slope in terms of its coefficients?

   Standard form is $Ax + By = C$. The slope is $m = -\frac{A}{B}$.

5. How do the slopes of parallel lines compare, and how do the slopes of perpendicular lines compare?

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

6. When solving a linear inequality, when must you flip the inequality sign?

   You flip the inequality sign whenever you multiply or divide both sides by a negative number.

7. How is the boundary line drawn when graphing $y > mx + b$ versus $y \geq mx + b$?

   For strict inequalities ($>$ or $<$) the boundary line is dashed; for inclusive inequalities ($\geq$ or $\leq$) it is solid. Shade the region satisfying the inequality.

8. What does it mean for a relation to be a linear function, and what shape is its graph?

   A linear function has a constant rate of change and can be written $f(x) = mx + b$; its graph is a straight line.

9. For the linear function $f(x) = mx + b$, what is the interpretation of $m$ in a real-world context?

   $m$ is the constant rate of change: the amount $f(x)$ increases (or decreases) for each one-unit increase in $x$.

10. What are the three possible numbers of solutions for a system of two linear equations, and what does each mean graphically?

    One solution (lines intersect at one point), no solution (parallel lines, never intersect), or infinitely many solutions (the same line).

11. How can you tell algebraically that a system of two linear equations has no solution?

    The variables have proportional coefficients but the constants are not in that same proportion, giving parallel lines with equal slopes and different $y$-intercepts (a false statement like $0 = 5$ results).

12. Describe the elimination method for solving a system of linear equations.

    Multiply equations so one variable has opposite coefficients, add the equations to eliminate that variable, solve for the remaining variable, then back-substitute.

13. What is a proportion, and how do you solve it using cross-multiplication?

    A proportion states two ratios are equal: $\frac{a}{b} = \frac{c}{d}$. Cross-multiply to get $a \cdot d = b \cdot c$, then solve.

14. What is the difference between a ratio and a rate?

    A ratio compares two quantities of the same kind (e.g., $3:2$), while a rate compares two quantities with different units (e.g., $60$ miles per hour).

15. What is the formula for percent change, and how do you distinguish an increase from a decrease?

    $$\text{Percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$$ A positive result is an increase; a negative result is a decrease.

16. How do you find the original value if a quantity of $120$ represents the result after a $20\%$ increase?

    Divide by the growth factor: $\frac{120}{1.20} = 100$. In general, original $= \frac{\text{final}}{1 + r}$ for an increase of rate $r$.

17. How do you express "$p$ percent of a number $N$" mathematically?

    $p\%$ of $N$ equals $\frac{p}{100} \times N$.

18. To convert a speed from meters per second to kilometers per hour, what conversion factor do you use?

    Multiply by $3.6$, since $1\ \text{m/s} = \frac{3600}{1000}\ \text{km/h} = 3.6\ \text{km/h}$.

19. When performing unit conversions with dimensional analysis, how are conversion factors arranged?

    Arrange each conversion factor as a fraction so unwanted units cancel diagonally, leaving only the desired unit in the final answer.

20. In a two-way frequency table, how do you compute a conditional relative frequency for a row?

    Divide each cell in the row by that row's total, so the relative frequencies represent proportions within that specific category.

21. On a scatterplot, what does a line of best fit represent, and what does its slope indicate?

    The line of best fit models the trend of the data; its slope indicates the predicted change in the $y$-variable per one-unit change in the $x$-variable.

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