This SAT Math Complete Reference covers the major skills students need for the Digital SAT Math section. It brings together algebra, functions, geometry, data analysis, probability, and key calculator strategies in one study sheet. Students need this cheat sheet to review common formulas quickly and recognize which method fits each problem type.
It is designed for grades 9-12 as a compact reference before practice tests and exam day.
The most important ideas include solving equations efficiently, interpreting graphs, using function notation, applying geometry formulas, and analyzing data displays. Students should know linear, quadratic, exponential, and proportional relationships, along with area, volume, and right triangle rules. Many SAT questions reward setting up the correct expression rather than doing long calculations.
Clear attention to units, answer choices, and restrictions helps prevent avoidable mistakes.
Key Facts
- Slope is , and a line in slope-intercept form is .
- Point-slope form is , which is useful when a line gives one point and a slope.
- A quadratic in standard form is , and its solutions are .
- The discriminant tells the number of real solutions: positive means , zero means , and negative means .
- For a right triangle, the Pythagorean theorem is , where is the hypotenuse.
- Circle area is , circumference is , and an arc length is .
- Percent change is , with increases positive and decreases negative.
- Probability is when all outcomes are equally likely.
Vocabulary
- Linear function
- A function with a constant rate of change that can be written as .
- Quadratic function
- A function that can be written as , where .
- System of equations
- A set of two or more equations that must be true at the same time.
- Vertex
- The highest or lowest point of a parabola, often written as in .
- Median
- The middle value of a data set when the values are arranged from least to greatest.
- Standard deviation
- A measure of how spread out data values are from the mean.
Common Mistakes to Avoid
- Confusing slope with the -intercept is wrong because gives the rate of change while gives the value of when .
- Forgetting parentheses when substituting negative numbers is wrong because but .
- Using diameter instead of radius in circle formulas is wrong because and require the radius , not the diameter.
- Dividing by a variable expression without checking restrictions is wrong because the expression may equal , which can remove valid cases or create invalid steps.
- Assuming correlation proves causation is wrong because two variables can be related without one directly causing the other.
Practice Questions
- 1 A line passes through and . Find its slope and write the equation in the form .
- 2 Solve using factoring or the quadratic formula.
- 3 A circle has radius . Find its area and circumference in terms of .
- 4 A scatterplot shows that as study time increases, test scores generally increase. Explain why this trend alone does not prove that study time caused the higher scores.
Understanding SAT Math Complete Reference
A strong SAT solution starts before any calculation. Turn each sentence into a mathematical fact. Name unknown quantities clearly, then write relationships between them.
Pay close attention to words such as at least, no more than, integer, positive, and distinct. These words can limit the possible answers. Units carry meaning too.
A rate in miles per hour cannot be combined directly with a time measured in minutes. In word problems, estimate what a sensible answer should look like before trusting a calculation. A negative number of people or an impossible percent often reveals an incorrect setup.
Functions appear in several forms, including equations, tables, graphs, and descriptions. The important skill is connecting these forms. A table may show a constant difference, which suggests a linear pattern.
A constant multiplier suggests exponential change. On a graph, intercepts show where an output is zero or where the graph meets an axis. The vertex of a parabola represents a highest or lowest output.
Learn to interpret what each feature means in context. For example, a zero might represent the time when profit becomes zero, not merely a point on a graph. Function notation is a compact way to describe an input and its output, so substitute carefully and keep parentheses when the input contains an expression.
Data questions often test reasoning more than arithmetic. The mean can change greatly when one value is unusually large or small. The median is often more stable in that situation.
A survey result is trustworthy only when the sample represents the population being discussed. A study can show an association between two variables without proving that one causes the other. When comparing percentages, identify the starting amount.
An increase of fifty percent followed by a decrease of fifty percent does not return to the original amount because the second change uses a different base. In probability, decide whether one event changes the chance of another. Events selected without replacement are dependent, so the total number of possible outcomes changes after each selection.
Geometry problems become easier when a diagram has labels, even if the test already provides a picture. Mark known lengths, angle relationships, parallel lines, and any right angles. Diagrams are not always drawn to scale, so rely on stated information rather than appearance.
Similar triangles are especially useful because matching angles create proportional side lengths. In coordinate geometry, distance, midpoint, slope, and area connect algebra to shapes. For three dimensional figures, separate the surface area from the volume.
Surface area measures covering material, while volume measures space inside a solid. Convert units before using a formula, since a length conversion affects square units and cubic units differently.
On the digital test, use the calculator to support thinking rather than replace it. Graphing can check roots, intersections, and the shape of a function. A table can test whether an answer fits a pattern.
Still, calculator output needs interpretation. Check the viewing window, rounding, and whether a displayed decimal should be an exact value. For multiple choice questions, plugging in answer choices can save time when the choices are numbers.
For grid in questions, verify format and signs carefully. Keep a steady routine of reading, modeling, solving, and checking. This reduces errors even when the underlying math is familiar.