The Heart of Algebra formulas center predominantly on the linear relationship, which is the most frequently tested concept on the PSAT. The foundational expression is the Slope-Intercept Form, written as $y = mx + b$. Here, $m$ represents the slope (rate of change) and $b$ represents the y-intercept (the initial value when $x=0$). To calculate the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$, students must use the Slope Formula: $m = (y_2 - y_1) / (x_2 - x_1)$. On the exam, you may encounter the Point-Slope Form, $y - y_1 = m(x - x_1)$, which is often more efficient when you are given a specific coordinate and the rate of change but lack the intercept. Understanding the Standard Form ($Ax + By = C$) is equally vital, as the PSAT often presents equations in this format to test your ability to quickly identify the slope as $-A/B$ and the y-intercept as $C/B$. Mastery of these forms allows for rapid graphing and interpretation of linear models.
Systems of Equations and Inequalities
Systems of equations require finding the point of intersection between two or more lines, representing the values that satisfy all equations simultaneously. The PSAT assesses this through three primary methods: substitution, elimination, and graphing. A critical concept for high-scoring students is the relationship between the number of solutions and the equations' coefficients. If two lines have the same slope but different y-intercepts, they are parallel and have no solution. If they have the same slope and the same y-intercept, they are the same line and have infinitely many solutions. For inequalities, the logic remains similar, but the solution set is a shaded region. You must remember that multiplying or dividing an inequality by a negative number requires flipping the inequality sign. This rule is a common trap in the PSAT math topics list, often placed in the final, more difficult questions of a module to catch students who are rushing.
Interpreting Linear Functions from Word Problems
The PSAT rarely asks you to solve an equation in a vacuum; instead, it embeds these formulas in real-world contexts. To excel, you must translate verbal phrases into mathematical symbols. Terms like "per," "each," or "rate" typically signify the slope ($m$), while "flat fee," "starting amount," or "initial value" signify the y-intercept ($b$). In a word problem, the independent variable ($x$) usually represents time or quantity, while the dependent variable ($y$) represents the total cost or result. One specific exam-style challenge involves determining the meaning of a constant within a provided model. For instance, in the equation $C = 15h + 40$, the PSAT might ask what the "40" represents. Recognizing this as the y-intercept allows you to identify it as the fixed cost before any hourly work ($h$) begins. This conceptual understanding is just as important as the calculation itself for maximizing your score.
Key Geometry and Trigonometry Concepts
Although the digital exam provides a Reference icon containing some PSAT geometry formulas, relying on it too heavily can waste precious seconds. You should have the area formulas for circles ($A = pi r^2$), rectangles ($A = lw$), and triangles ($A = 1/2 bh$) memorized. Furthermore, understand the Circumference of a circle ($C = 2pi r$ or $C = pi d$) and how it relates to arc length. The PSAT often tests the Arc Length Formula: $s = (n/360) \times 2pi r$, where $n$ is the central angle in degrees. For three-dimensional shapes, focus on the volume of rectangular prisms ($V = lwh$) and right cylinders ($V = pi r^2 h$). A common exam tactic is to provide the diameter when the formula requires the radius, or to ask for the volume in different units than those provided in the prompt, necessitating a unit conversion before applying the formula.
Triangle Rules and the Pythagorean Theorem
Triangles are a cornerstone of the PSAT geometry section. The Pythagorean Theorem ($a^2 + b^2 = c^2$) is essential for solving right triangle problems, but savvy test-takers also memorize Pythagorean Triples like 3-4-5 and 5-12-13 to save time. Beyond basic right triangles, you must master Special Right Triangles: the 45-45-90 triangle (with side ratios $x : x : xsqrt{2}$) and the 30-60-90 triangle (with side ratios $x : xsqrt{3} : 2x$). The exam also frequently tests the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This rule is often used in "find the possible value" questions. Additionally, remember that the sum of interior angles in any triangle is always 180 degrees, a fact that is frequently combined with parallel line theorems (alternate interior angles) in complex diagrams.
Basic Right Triangle Trigonometry (SOH CAH TOA)
Trigonometry on the PSAT is limited but high-impact. You must know the definitions of Sine, Cosine, and Tangent based on the mnemonic SOH CAH TOA: $sin = \text{opposite/hypotenuse}$, $cos = \text{adjacent/hypotenuse}$, and $\tan = \text{opposite/adjacent}$. A crucial relationship frequently tested is the Complementary Angle Relationship: $sin(x) = cos(90 - x)$. This means that the sine of one acute angle in a right triangle is equal to the cosine of the other acute angle. You might encounter a question that provides $sin(a) = 4/5$ and asks for the value of $cos(b)$ in the same right triangle; knowing this identity allows you to answer instantly without further calculation. While you won't need the Law of Sines or Law of Cosines, a firm grasp of these basic ratios and their relationships to the unit circle is vital for the Geometry and Trigonometry domain.
Passport to Advanced Math: Beyond Algebra
The Passport to Advanced Math equations move into the realm of non-linear functions. The Quadratic Formula, $x = [-b ± sqrt{b^2 - 4ac}] / 2a$, is your primary tool for finding the roots of a quadratic equation in the form $ax^2 + bx + c = 0$. Within this formula lies the discriminant ($b^2 - 4ac$), which determines the number of real solutions: if positive, there are two; if zero, one; if negative, none. You must also be proficient in finding the Vertex of a parabola. For an equation in standard form, the x-coordinate of the vertex is $h = -b/2a$. For an equation in Vertex Form, $y = a(x - h)^2 + k$, the vertex is simply $(h, k)$. Polynomial operations, including long division or the Remainder Theorem, may also appear. The Remainder Theorem states that for a polynomial $f(x)$, the remainder of the division by $(x - a)$ is simply $f(a)$, a shortcut that saves significant time on multiple-choice questions.
Laws of Exponents and Radical Expressions
Handling exponents and radicals efficiently is a hallmark of an advanced PSAT candidate. You must be fluent in the Product Rule ($x^a cdot x^b = x^{a+b}$), the Quotient Rule ($x^a / x^b = x^{a-b}$), and the Power Rule ($(x^a)^b = x^{ab}$). Perhaps most important for the PSAT is the relationship between radicals and rational exponents: $sqrt[n]{x^m} = x^{m/n}$. This allows you to convert complex radical expressions into exponential form, making them easier to simplify using the standard laws. Negative exponents are another frequent feature; remember that $x^{-n} = 1/x^n$. The exam often combines these rules, requiring you to simplify a multi-variable expression before solving for a specific value. Precision here is key, as small errors in sign or arithmetic are the most common reasons for incorrect answers in this category.
Graphing Nonlinear Functions
Understanding the visual representation of functions is essential for the PSAT. For quadratics, the coefficient $a$ determines if the parabola opens upward ($a > 0$) or downward ($a < 0$). For Exponential Functions, represented by $y = ab^x$, the variable $a$ is the y-intercept (initial value) and $b$ is the growth or decay factor. If $b > 1$, the function shows growth; if $0 < b < 1$, it shows decay. The PSAT also tests your ability to identify asymptotes and the behavior of functions as $x$ approaches infinity. You should be able to recognize the graphs of absolute value functions ($v$-shaped) and circle equations on the coordinate plane: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius. Being able to quickly identify the center and radius from an equation—sometimes requiring you to complete the square first—is a high-level skill that appears in the latter half of the math modules.
Ratios, Proportions, and Percentages
Problem Solving and Data Analysis relies heavily on proportional reasoning. A Proportion is an equation stating that two ratios are equal ($a/b = c/d$), which is often solved via cross-multiplication. For Percentage Change, use the formula: $[(\text{New Value} - ext{Old Value}) / ext{Old Value}] imes 100$. It is vital to remember that the denominator is always the original value, not the new one. In problems involving multiple percentage changes, such as a discount followed by a tax, do not simply add the percentages. Instead, multiply the original amount by the respective growth or decay factors. For example, a 20% discount is a multiplier of 0.80, and a 5% tax is a multiplier of 1.05. This domain also includes unit conversions, where you must use the Factor-Label Method to ensure units cancel out correctly, leaving you with the desired dimension in the final answer.
Statistical literacy on the PSAT involves both calculation and interpretation. The Arithmetic Mean is the sum of all values divided by the number of values ($n$). The Median is the middle value when the data set is ordered from least to greatest. If the data set has an even number of values, the median is the average of the two middle terms. While you rarely have to calculate Standard Deviation by hand, you must understand it conceptually as a measure of how spread out the data points are from the mean. A higher standard deviation indicates a wider spread. The PSAT often asks how adding or removing an outlier (a value significantly different from the rest) affects the mean and median. The mean is highly sensitive to outliers, while the median is more robust and typically changes very little. Understanding this distinction is key to answering conceptual statistics questions correctly.
Interpreting Data from Tables and Graphs
A significant portion of the PSAT involves extracting information from visual data. You must be comfortable with Conditional Probability, which is calculated as $P(A|B) = P(A ext{ and } B) / P(B)$. In a two-way table, this means narrowing your focus to a specific row or column (the denominator) rather than the entire sample size. When analyzing Scatterplots, look for the Line of Best Fit. The PSAT may ask you to identify the equation of this line or to interpret the meaning of its slope in context. Be wary of the difference between correlation and causation; a strong linear relationship on a graph does not prove that one variable causes the other. Additionally, understand how to read box-and-whisker plots, focusing on the Interquartile Range (IQR), which is the distance between the first quartile (25th percentile) and the third quartile (75th percentile).
The most common obstacle for students is not forgetting a formula, but failing to recognize which one to apply. This skill, often called problem categorization, involves looking for keywords and structural cues. If a question mentions "zeros," "roots," or "x-intercepts" of a curved line, you should immediately think of the quadratic formula or factoring. If a problem describes a constant rate of change, it is a linear equation problem. To improve at this, practice using your must-know PSAT math formulas on a variety of problem types. When you encounter a word problem, first identify what the question is asking for (e.g., a total amount, a rate, or a specific coordinate). Then, list the known variables. Matching these knowns to the variables in your formula library will clarify the path to the solution, reducing the "blank page" anxiety often felt during the exam.
Errors on the PSAT math section often stem from misapplying a memorized formula rather than total ignorance. One frequent mistake is failing to convert the Equation of a Circle into the standard form $(x - h)^2 + (y - k)^2 = r^2$ before identifying the radius. Students often see the constant on the right side and forget it represents $r^2$, not $r$. Another pitfall occurs in Exponent Rules, specifically confusing $x^a + x^a$ (which is $2x^a$) with $x^a cdot x^a$ (which is $x^{2a}$). In geometry, students often forget that the area of a triangle formula requires the perpendicular height, not the length of a slanted side. To avoid these traps, always write out the formula in its general form before plugging in numbers. This physical act of writing the formula helps engage your motor memory and provides a visual check against common substitution errors.
To solidify your understanding of this PSAT equation sheet content, you must engage in targeted practice. For the Heart of Algebra, solve problems where you must find the intersection of two lines using elimination, then verify the result by substituting the values back into both original equations. For the Passport to Advanced Math, practice completing the square for a quadratic equation to find its vertex. In the Problem Solving domain, use real-world data sets to calculate the mean and median, then observe how the mean shifts when you add a value much larger than the others. Finally, for Geometry, practice finding the volume of composite shapes—such as a cylinder topped with a hemisphere—which requires you to combine multiple volume formulas. Consistent application across these categories ensures that by test day, these formulas are not just memorized strings of characters, but active tools in your problem-solving arsenal.