The Heart of Algebra domain focuses heavily on linear relationships, which account for approximately 33% of the Math section. The most fundamental tool here is the Slope-Intercept Form, written as $y = mx + b$. In this equation, $m$ represents the slope (rate of change) and $b$ represents the y-intercept (initial value). Students must be able to calculate the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ using the formula $m = (y_2 - y_1) / (x_2 - x_1)$. Beyond simple calculation, the SAT often tests the conceptual meaning of these components in real-world contexts, such as a taxi fare where the y-intercept is the base fee and the slope is the cost per mile. Additionally, understanding the Point-Slope Form, $y - y_1 = m(x - x_1)$, allows for rapid equation construction when given a single point and a rate, bypassing the need to solve for $b$ first.
Systems of Equations & Inequalities
When dealing with systems of equations, the SAT assesses your ability to find the intersection of two lines or determine the number of solutions. A system has no solution if the lines are parallel, meaning they have the exact same slope but different y-intercepts. Conversely, a system has infinitely many solutions if the equations represent the same line, sharing both slope and y-intercept. For inequalities, the Solution Set is represented by the shaded region on a coordinate plane. A critical rule to remember is that when you multiply or divide an inequality by a negative number, you must flip the inequality sign. This is a common trap in multi-step algebra problems where isolated variables are negative. Mastering the substitution and elimination methods is essential for speed, as the SAT timing allows for less than 90 seconds per question on average.
Exponential Growth and Decay Models
While linear equations change by a constant amount, exponential models change by a constant percentage. The standard formula is $y = a(1 ± r)^t$, where $a$ is the initial amount, $r$ is the rate of growth or decay expressed as a decimal, and $t$ is time. This is a subset of the SAT Heart of Algebra formulas that frequently appears in word problems involving population growth or compound interest. If the value inside the parentheses is greater than 1, the function represents growth; if it is between 0 and 1, it represents decay. It is vital to distinguish between the "growth factor" $(1+r)$ and the "growth rate" $(r)$. For instance, a 15% increase results in a growth factor of 1.15. Misinterpreting these terms often leads to incorrect answer choices designed to catch students who confuse the percentage with the multiplier.
Passport to Advanced Math: Essential Equations
The Passport to Advanced Math equations require a deeper understanding of non-linear structures. The Quadratic Formula, $x = (-b ± √(b^2 - 4ac)) / 2a$, is indispensable for finding the roots of a quadratic equation in standard form ($ax^2 + bx + c = 0$). Within this formula lies the Discriminant ($b^2 - 4ac$), which determines the nature of the solutions: a positive discriminant signifies two real solutions, zero signifies one real solution, and a negative value indicates two complex solutions. Furthermore, the SAT frequently asks for the sum or product of the roots. Instead of solving the entire quadratic, savvy test-takers use Vieta’s Formulas: the sum of the roots is $-b/a$ and the product is $c/a$. These shortcuts are high-leverage tools that save significant time during the No-Calculator section.
Function notation, $f(x)$, is simply a way to relate an input to an output. The SAT tests your ability to manipulate these functions through Transformations. If given $f(x)$, then $f(x) + k$ shifts the graph up by $k$ units, while $f(x - h)$ shifts the graph to the right by $h$ units. Note the inverse relationship in horizontal shifts: a minus sign moves the graph in the positive x-direction. Reflections are also common; $-f(x)$ reflects the graph across the x-axis, while $f(-x)$ reflects it across the y-axis. Understanding the Vertex Form of a quadratic, $y = a(x - h)^2 + k$, is particularly useful because it explicitly states the vertex $(h, k)$ of the parabola. This allows for immediate identification of the maximum or minimum value of the function without performing completing the square or differentiation.
Rules of Exponents and Radicals
Efficiency in the Advanced Math section depends on the fluid application of exponent laws. Key rules include the Product Rule ($x^a ⋅ x^b = x^{a+b}$), the Power Rule ($(x^a)^b = x^{ab}$), and the Negative Exponent Rule ($x^{-a} = 1/x^a$). Perhaps most important for the SAT is the relationship between radicals and rational exponents: $∛x^a = x^{a/b}$. This conversion is frequently required to simplify expressions or solve equations where the variable is in the exponent. Students should also be familiar with the Conjugate method for simplifying expressions with radicals or complex numbers in the denominator. For example, to simplify $1 / (3 + √2)$, you multiply the numerator and denominator by $(3 - √2)$. This eliminates the radical from the denominator, a necessary step for matching your answer to the multiple-choice options provided.
Data analysis questions often focus on measures of central tendency and spread. The Arithmetic Mean is the sum of all values divided by the number of values. A common SAT variation is the "Weighted Average" or finding a missing value given a target mean. For instance, if four test scores are known, what must the fifth score be to reach an average of 90? The Median is the middle value when data is ordered numerically; if there is an even number of values, it is the average of the two middle terms. The SAT also tests Standard Deviation conceptually. You won't be asked to calculate it using a complex formula, but you must know that it measures the "spread" of data. A data set where the values are close to the mean has a lower standard deviation than one where the values are widely dispersed.
Rates, Ratios, Proportions, and Percentages
This section of the exam evaluates your ability to scale quantities and convert units. The Distance Formula ($d = rt$) is a staple, but it applies to any rate (e.g., Work = Rate × Time). When dealing with Proportions, the cross-multiplication method is the most reliable way to solve for an unknown variable. Percent change is another high-frequency topic, calculated as (New - Old) / Old × 100. A common pitfall is applying a percentage increase and then a percentage decrease of the same value; students must remember that these do not cancel out because the base value changes after the first operation. For example, a 10% increase followed by a 10% decrease results in a net decrease of 1% because the second calculation is performed on 110% of the original amount.
Interpreting Data from Tables and Graphs
Success in Data Analysis requires more than just a must-know SAT math formulas list; it requires literacy in visual data. Conditional Probability is frequently tested using two-way tables. This is the probability of an event occurring given that another condition is already met. The formula is $P(A|B) = P(A ∩ B) / P(B)$. In simple terms, you must restrict the denominator to the specific row or column defined by the condition. Additionally, understanding Line of Best Fit in scatterplots is crucial. The SAT will ask you to identify the equation that most closely models the data or to interpret a specific data point's Residual, which is the vertical distance between the actual observed value and the value predicted by the model.
Additional Topics in Math: Geometry & Trigonometry
While the SAT provides a math reference sheet for geometry, speed is your greatest asset. You should have the volume of a cylinder ($V = πr^2h$) and the volume of a rectangular prism ($V = lwh$) memorized to avoid flipping back to the first page. A critical concept not always detailed on the reference sheet is the relationship between similar figures. If the ratio of the side lengths of two similar shapes is $a:b$, then the ratio of their areas is $a^2:b^2$, and the ratio of their volumes is $a^3:b^3$. This Scaling Factor rule is a shortcut for complex geometry problems that would otherwise require multiple steps of calculation. Knowing the Sum of Interior Angles formula for a polygon, $(n-2) × 180$, where $n$ is the number of sides, is also vital for solving multi-sided figure problems.
Circle Theorems and Sector Calculations
Circles on the SAT go beyond just area ($πr^2$) and circumference ($2πr$). You must know the Equation of a Circle in the coordinate plane: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius. If an equation is given in expanded form, you must use completing the square to return it to this standard form. Furthermore, the exam tests Arc Length and Sector Area. These are fractional parts of the whole circle, calculated using the ratio of the central angle to 360 degrees. For instance, Arc Length $= (central angle / 360) ⋅ 2πr$. Understanding the relationship between degrees and Radians is also essential; to convert degrees to radians, multiply by $π/180$. A full circle is $2π$ radians, and a straight line is $π$ radians.
Basic Trigonometry (SOH CAH TOA) and the Pythagorean Theorem
Right triangle trigonometry is centered on the SOH CAH TOA mnemonic: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, and Tangent = Opposite/Adjacent. A sophisticated concept often tested is the Complementary Angle Relationship: $\sin(x) = \cos(90 - x)$. This means the sine of one acute angle in a right triangle is equal to the cosine of the other. The Pythagorean Theorem ($a^2 + b^2 = c^2$) is the foundation for finding missing side lengths, and students should recognize Pythagorean Triples like 3-4-5 and 5-12-13 to save time. Additionally, the SAT expects familiarity with Special Right Triangles: the 45-45-90 triangle (sides $x, x, x√2$) and the 30-60-90 triangle (sides $x, x√3, 2x$). Recognizing these ratios allows you to solve geometry problems without needing a calculator or complex trigonometric functions.
Active Recall vs. Passive Review Techniques
Simply reading an SAT math cheat sheet is a passive activity that rarely leads to long-term retention. To truly master these formulas, you must use Active Recall. This involves testing yourself without looking at the answers. Cover the right side of your formula list and try to write down the equation for the vertex of a parabola or the area of a sector from memory. Research shows that the act of retrieving information strengthens neural pathways more effectively than repeated exposure. Another powerful technique is Feynman Technique, where you explain the logic of a formula to someone else. If you can explain why the discriminant tells us the number of solutions, you are far more likely to remember the formula itself during the high-pressure environment of the testing center.
Contextual Learning Through Practice Problems
Formula memorization is useless without the ability to recognize Problem Cues. Each formula on the SAT is linked to specific keywords or visual patterns. For example, when a question mentions "minimum value" or "maximum value" of a quadratic, your brain should immediately trigger the Vertex Formula. When you see a percentage increase over time, you should reach for the exponential growth model. The best way to build these associations is through targeted practice. Instead of doing random problems, group your practice by domain. Spend an entire session on Heart of Algebra to see how the same SAT math formulas are used across different question formats. This contextual learning ensures that you don't just know the formula, but you know exactly when to deploy it.
While using a pre-made SAT formula sheet is a good starting point, creating your own is a superior study method. Start with a blank page and categorize the formulas by the four SAT math domains. As you take practice tests, identify which formulas you struggled to remember or which ones you used most frequently. Add these to your personalized sheet along with small diagrams or "watch out" notes for common mistakes you make. This document becomes a living record of your mathematical growth. By the week of the exam, your personal sheet should be internalized. The goal is to reach a level of Automaticity, where the formula appears in your mind the moment you finish reading the prompt, allowing you to focus your mental energy on the logic of the problem rather than the mechanics of the math.
The Quadratic Formula is one of the most complex strings of characters you need to memorize, and it is easy to make a small but lethal error. The most common mistake is forgetting that the entire expression $(-b ± √(b^2 - 4ac))$ is over $2a$. Many students accidentally only put the radical over $2a$. Another frequent error occurs with the Sign of the Coefficients. If $b$ is already negative, $-b$ becomes positive. Similarly, if $c$ is negative, the $-4ac$ term becomes positive because you are subtracting a negative. To mitigate these risks, always write out the values of $a, b,$ and $c$ explicitly on your scratch paper before plugging them into the formula. This small step reduces the chance of a sign error that could lead to one of the "distractor" answer choices.
The SAT often presents problems where two formulas look similar but serve different purposes. A classic example is the confusion between the Area of a Circle ($πr^2$) and the Circumference of a Circle ($2πr$). Because both involve $π, r,$ and the number 2, students under time pressure often swap them. Another common mix-up occurs with the Slope Formula and the Midpoint Formula. The slope involves subtraction and division ($y_2 - y_1$), while the midpoint involves addition and division ($(x_1 + x_2)/2$). To avoid these errors, focus on the units: area is always in square units ($r^2$), while circumference and slope are linear. Midpoint is a coordinate pair, not a single value. Checking the "dimensionality" of your formula can serve as a quick sanity check during the exam.
Forgetting Units in Geometric Calculations
One of the most frustrating ways to lose points is by performing the correct calculation but failing to account for Unit Conversion. The SAT frequently provides dimensions in one unit (e.g., feet) but asks for the answer in another (e.g., inches or square yards). This is especially prevalent in Volume problems. Remember that if you are converting square units, you must square the conversion factor. For example, since there are 3 feet in a yard, there are $3^2 = 9$ square feet in a square yard. If you are converting cubic units, you must cube the factor ($3^3 = 27$ cubic feet in a cubic yard). Always circle the required units in the question stem before you begin your calculations. This provides a visual reminder to perform the final conversion step before selecting your answer, ensuring your hard work results in a correct score.