Quadratic equations are a cornerstone of the ACT, appearing frequently in both straightforward algebraic manipulation and complex word problems. The Quadratic Formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, is your primary tool when a trinomial cannot be easily factored. Beyond the formula itself, the discriminant ($b^2 - 4ac$) is a vital sub-concept. If the discriminant is positive, the equation has two real roots; if zero, one real root; and if negative, two complex roots. The ACT often tests this by asking how many times a parabola intersects the x-axis. Furthermore, mastering the Difference of Squares ($a^2 - b^2 = (a - b)(a + b)$) and perfect square trinomials allows for rapid factoring without the need for the full formula, saving precious time during the first 30 questions where speed is paramount.
Linear Equations and Slope
Linear relationships dominate the early to middle sections of the exam. The Slope-Intercept Form, $y = mx + b$, is the most common representation, where $m$ is the slope and $b$ is the y-intercept. To find the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$, you must use the Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$. The ACT frequently tests your understanding of parallel and perpendicular lines. Parallel lines share the same slope, while perpendicular lines have slopes that are negative reciprocals (e.g., $1/2$ and $-2$). If you encounter a linear equation in Standard Form ($Ax + By = C$), remember that the slope is calculated as $-A/B$. Recognizing these patterns instantly prevents calculation errors when rearranging equations under time constraints.
Exponent and Radical Rules
Exponent properties are frequently tested through simplification problems. 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 ($(x^a)^b = x^{ab}$). A common trap on the ACT involves negative exponents; remember that $x^{-n} = 1/x^n$. Additionally, the relationship between radicals and fractional exponents is essential: $\sqrt[n]{x^a} = x^{a/n}$. You may see questions that require you to rationalize a denominator or simplify a square root by finding the largest perfect square factor. Understanding that $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ allows you to break down complex radicals into manageable parts, which is a common requirement for late-section questions involving higher-level algebra.
Percentages and Ratios
Basic arithmetic on the ACT often masks itself in multi-step percentage and ratio problems. The fundamental formula for Percent Change is $\frac{\text{New} - \text{Old}}{\text{Old}} \times 100$. Many students mistakenly divide by the new value, which leads to one of the common distractor answers. For ratio problems, the Ratio Sum Method is highly effective: if the ratio of two quantities is $3:5$, the total can be represented as $3x + 5x = \text{Total}$. This allows you to solve for the constant $x$ and find the individual parts. You must also be comfortable with proportions, where two ratios are set equal to each other ($a/b = c/d$), requiring cross-multiplication to find the missing variable. These concepts are frequently applied to "real-world" scenarios like tax, tips, and scale drawings.
Geometry constitutes roughly 20-25% of the ACT math content, and area calculations are foundational. For any triangle, the area is $A = \frac{1}{2}bh$, where the height must be perpendicular to the base. For circles, the Area of a Circle is $A = \pi r^2$. A frequent ACT variation involves finding the Area of a Sector, which uses the formula $(\frac{\theta}{360}) \pi r^2$, where $\theta$ is the central angle. For rectangles and parallelograms, $A = lw$ or $A = bh$ applies. The exam often presents "composite figures"—shapes made of multiple polygons—where you must calculate individual areas and add or subtract them. For example, finding the area of a shaded region between a square and an inscribed circle requires calculating both $s^2$ and $\pi r^2$ and subtracting the latter from the former.
Perimeter and Circumference
Perimeter is the sum of all outer sides of a polygon. For a circle, this distance is the Circumference, calculated as $C = 2\pi r$ or $C = \pi d$. The ACT often links circumference to the Arc Length of a circle using the ratio $(\frac{\theta}{360}) 2\pi r$. Understanding the relationship between the radius, diameter, and circumference is critical for solving problems where only one piece of information is provided. In polygon problems, watch for "fencing" or "border" word problems, which are coded requests for perimeter. It is also worth noting that for any regular polygon, the perimeter is simply $ns$, where $n$ is the number of sides and $s$ is the side length. Mastery of these formulas for ACT math ensures you don't stall on the simpler geometric questions.
Properties of Special Right Triangles
While the Pythagorean Theorem is useful, the ACT rewards students who recognize Special Right Triangles because they allow for much faster calculations. The two primary types are the 45-45-90 triangle and the 30-60-90 triangle. In a 45-45-90 triangle, the sides follow the ratio $x : x : x\sqrt{2}$. In a 30-60-90 triangle, the ratio is $x : x\sqrt{3} : 2x$, where $x$ is the side opposite the 30-degree angle. These ratios are essential for solving problems involving squares (bisected by a diagonal) or equilateral triangles (bisected by an altitude). If a problem asks for the exact height of an equilateral triangle with side length 10, recognizing it as two 30-60-90 triangles allows you to instantly identify the height as $5\sqrt{3}$ without using the Pythagorean Theorem.
The Pythagorean Theorem
For any right triangle, the Pythagorean Theorem states that $a^2 + b^2 = c^2$, where $c$ is the hypotenuse. The ACT frequently uses Pythagorean Triples—sets of integers that satisfy the theorem—to reward prepared students. The most common triples are $(3, 4, 5)$, $(5, 12, 13)$, $(8, 15, 17)$, and $(7, 24, 25)$, along with their multiples (e.g., $6, 8, 10$). Recognizing these triples can save 30 to 45 seconds of calculation time per question. Additionally, you may need the Distance Formula in coordinate geometry, which is simply a variation of the Pythagorean Theorem: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Understanding that the distance between two points is actually the hypotenuse of a right triangle helps in visualizing complex coordinate geometry problems.
Volume of Prisms, Cylinders, and Spheres
Three-dimensional geometry questions usually focus on volume. The general rule for the Volume of a Prism or cylinder is $V = Bh$, where $B$ is the area of the base. For a rectangular prism, this is $V = lwh$; for a cylinder, $V = \pi r^2 h$. The ACT may also test the Volume of a Cone or Volume of a Pyramid, both of which are $V = \frac{1}{3} Bh$. While the Volume of a Sphere ($V = \frac{4}{3} \pi r^3$) is occasionally provided within the question text, you should memorize it to maintain momentum. Pay close attention to the units; if the base area is in square inches but the height is in feet, you must convert before applying the volume formula.
Surface Area Calculations
Surface area is the sum of the areas of all faces of a 3D object. For a rectangular prism, the Surface Area formula is $SA = 2lw + 2lh + 2wh$. For a cylinder, it includes the two circular bases and the "lateral area": $SA = 2\pi r^2 + 2\pi rh$. The ACT test-makers often create "open-top" box problems where you must remember to exclude one of the faces from your calculation. Unlike volume, which measures capacity, surface area measures coverage, making it common in word problems involving paint, wrapping paper, or metal plating. Precision is key here, as forgetting just one face of the object will lead to an incorrect answer choice specifically designed to catch that error.
Converting Units of Measurement
Unit conversion is a subtle but frequent source of error on the ACT. You must be comfortable with Dimensional Analysis, ensuring that units cancel out correctly. Common conversions include feet to inches, yards to feet, and hours to minutes. However, the ACT also tests squared and cubed unit conversions. For example, if $1 \text{ yard} = 3 \text{ feet}$, then $1 \text{ square yard} = 9 \text{ square feet}$ ($3^2$), and $1 \text{ cubic yard} = 27 \text{ cubic feet}$ ($3^3$). Many students fail to square or cube the conversion factor, leading to significant errors in geometry problems. Always check the final units requested in the question before selecting your answer, as the ACT often provides the pre-conversion result as a distractor.
Trigonometry and Coordinate Geometry
SOH CAH TOA and Right Triangle Trig
Trigonometry on the ACT is primarily based on right triangles and the SOH CAH TOA mnemonic. This stands for Sine (Opposite/Hypotenuse), Cosine (Adjacent/Hypotenuse), and Tangent (Opposite/Adjacent). You must be able to use these ratios to find missing side lengths or angles using inverse trigonometric functions. Additionally, you should know the Tangent Identity ($\tan \theta = \sin \theta / \cos \theta$) and the Pythagorean Identity ($\sin^2 \theta + \cos^2 \theta = 1$). While more advanced concepts like the Law of Sines or Law of Cosines appear rarely (usually in the final 10 questions), a firm grasp of basic sine, cosine, and tangent is essential for the middle-to-late difficulty brackets of the exam.
Equation of a Line and Slope
In coordinate geometry, the ACT math equations you need extend to the Point-Slope Form: $y - y_1 = m(x - x_1)$. This is often more efficient than slope-intercept form when you are given a point and a slope and need to find the equation quickly. You must also understand how to find the x-intercept (set $y=0$) and y-intercept (set $x=0$). The ACT also tests the relationship between slopes and the steepness of a line; a larger absolute value of $m$ indicates a steeper incline. If you are asked to find the intersection of two lines, you are essentially solving a system of linear equations, which can be done through substitution or elimination.
The Midpoint Formula is used to find the center point between $(x_1, y_1)$ and $(x_2, y_2)$ and is expressed as $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$. It is essentially the average of the x-coordinates and the average of the y-coordinates. The Distance Formula, as mentioned previously, is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. A common ACT problem provides one endpoint and the midpoint and asks you to find the other endpoint. In this case, it is often faster to use the "step method"—calculating the horizontal and vertical distance from the endpoint to the midpoint and then applying that same "step" to find the second endpoint—rather than plugging values into the formula and solving algebraically.
Graphing Basic Functions
You must recognize the standard forms and shapes of various functions. This includes Quadratic Functions ($y = ax^2 + bx + c$), which form parabolas, and Circle Equations in the coordinate plane: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius. The ACT also tests Function Transformations. For any function $f(x)$, $f(x) + k$ shifts the graph up, $f(x - h)$ shifts it to the right, and $-f(x)$ reflects it across the x-axis. Being able to visualize these shifts allows you to eliminate incorrect graph options quickly without having to plot individual points, a technique that is vital for the final third of the math section.
Statistics, Probability, and Data Analysis
Statistics questions on the ACT focus on Measures of Central Tendency. The Mean (average) is the $\text{Sum of terms} / \text{Number of terms}$. A common ACT "Average" problem involves adding a new value to a set and calculating the new mean, which is best handled using the Sum Formula: $\text{Sum} = \text{Average} \times \text{Number of terms}$. The Median is the middle value when the data is ordered; for an even number of terms, it is the average of the two middle values. The Mode is the most frequent value, and the Range is the difference between the maximum and minimum values. Understanding how an outlier (a value significantly higher or lower than the rest) affects these measures is a frequent conceptual question; the mean is heavily affected by outliers, while the median remains relatively stable.
Basic Probability Calculations
Probability on the ACT is defined as the number of favorable outcomes divided by the total number of possible outcomes. For multiple independent events, you use the Multiplication Rule: the probability of both Event A and Event B occurring is $P(A) \times P(B)$. If the events are mutually exclusive, the probability of either occurring is $P(A) + P(B)$. The ACT also includes "without replacement" problems, where the total number of outcomes decreases after the first event (e.g., drawing two red marbles from a bag without putting the first one back). These problems require careful adjustment of the denominator for the second fraction in your multiplication. Expected value, another probability concept, is calculated by multiplying each outcome by its probability and summing the results.
Interpreting Graphs and Charts
Data analysis involves extracting information from bar graphs, line graphs, scatterplots, and tables. You must be able to identify trends, such as positive correlation (as $x$ increases, $y$ increases) and negative correlation. The ACT often asks you to find the Line of Best Fit on a scatterplot, which is the linear equation that best represents the data trend. Pay attention to the axes; sometimes they do not start at zero, or they use different scales, which can make a small change look significant. You may also encounter Frequency Tables, where you must calculate the mean or median based on how often each value occurs. Success here depends on "reading the fine print" of the graph's legend and labels to ensure you are using the correct data points for your formulas.
Translating Word Problems into Equations
The bridge between knowing a formula and scoring well is the ability to translate English into math. Keywords are your guide: "is" means equals, "of" means multiply, and "per" indicates a rate or division. For example, a problem stating "the cost is five dollars more than twice the number of items" translates to $C = 2n + 5$. Many must-know ACT formulas are applied within these translations, particularly those involving distance, rate, and time ($d = rt$). When faced with a paragraph-long problem, your first step should be to identify what the question is asking for (the "target") and then list the known variables. This systematic approach prevents "information overload" and helps you identify which formula from your mental ACT math cheat sheet is required.
Multi-Step Problem Strategies
Harder ACT questions often require the use of two or more formulas in sequence. For instance, you might need to use the Pythagorean Theorem to find the height of a triangle before you can calculate its area, or use the midpoint formula to find the center of a circle before using the distance formula to find its radius. The key strategy for these problems is intermediate labeling. As you calculate each part, write down the value and what it represents. This prevents you from using a diameter when you need a radius, or a slope when you need a y-intercept. In the final 20 questions, expect these "chained" problems where the output of one equation becomes the input for the next.
Common Pitfalls and How to Avoid Them
One of the biggest pitfalls is the "partial answer" trap. The ACT often includes answer choices that correspond to intermediate steps. If a question asks for the area of a circle but you stop after finding the radius, your current number will likely be one of the choices. Another common error is failing to maintain unit consistency, such as mixing inches and feet. To avoid these, always re-read the final sentence of the question before bubbling in your answer. Additionally, be wary of "NOT" or "EXCEPT" questions, which require you to find the one formula or value that does not work. Marking your test booklet with your calculations allows you to double-check your logic if you have time at the end of the section.
While you cannot bring a cheat sheet into the testing center, the act of creating one is a powerful mnemonic tool. Organize your sheet by category: Algebra, Geometry, Trig, and Stats. Instead of just writing the formula, include a small diagram or a note about a common trap. For example, next to the area of a triangle, write "height must be perpendicular." This personalized ACT math cheat sheet should be your primary study document. Review it daily in the two weeks leading up to the exam. The goal is "fluency"—you should be able to recite the formula for the volume of a cylinder as quickly as you can say your own phone number.
Effective Memorization Techniques
To move formulas from short-term to long-term memory, use active recall and spaced repetition. Don't just read the list; cover the right side of the page and try to write the formula from memory. Use flashcards for the most difficult ones, like the quadratic formula or the circle equation. Another effective method is "derivation." If you understand that the area of a triangle is just half of a parallelogram, or that the distance formula is just the Pythagorean Theorem, you are less likely to forget them under pressure. Linking formulas to visual shapes or "stories" can also help in retaining the more abstract trigonometric identities.
Memorization is useless without application. Once you have a formula down, solve at least five targeted practice problems that require it. Use official ACT practice tests to see how the exam phrases questions. Notice how the test-makers often hide the need for a formula; for example, a problem about a "circular rug" is a prompt for circle area or circumference. As you practice, move away from looking at your notes. If you can't solve a problem without checking your formula sheet, you haven't fully mastered that concept yet. By the time you reach the actual exam, the application of these formulas should be instinctive, allowing you to focus your mental energy on the most challenging, logic-based questions at the end of the test.