SAT Math: Concepts, Methods, and Worked Problems

Slope

The rate of change of a line; it tells how much y changes for each 1-unit increase in x.

y-intercept

The value of $y$ when $x = 0$; in $y = mx + b$, it is $b$ and represents the starting value.

Slope-intercept form

The linear form y = mx + b, where m is the slope and b is the y-intercept.

Point-slope form

The linear form $y - y_1 = m(x - x_1)$, used when the slope $m$ and a point $(x_1, y_1)$ are known.

Slope formula

For points $(x_1, y_1)$ and $(x_2, y_2)$, $m = \frac{y_2 - y_1}{x_2 - x_1}$; the order must stay consistent.

Parallel lines

Lines with the same slope that never intersect if they are distinct.

Perpendicular lines

Lines that meet at a right angle; for nonvertical lines, their slopes are negative reciprocals.

Vertical line

A line with equation x = c; its slope is undefined.

Horizontal line

A line with equation y = c; its slope is 0.

Linear inequality

An inequality involving a linear expression, such as $2x + 3 \leq 11$; if you multiply or divide by a negative, you must flip the inequality sign.

System of linear equations

A pair or set of linear equations solved at the same time; graphically, the solution is the point where the lines intersect.

Substitution method

A system-solving method where one equation is solved for a variable and then substituted into the other equation.

Elimination method

A system-solving method where equations are added or subtracted to cancel a variable.

Function

A rule that assigns each input exactly one output.

Function notation

Notation like f(x), read as the output of function f for input x; it does not mean f times x.

Domain

The set of allowed inputs of a function, often limited by denominators, even roots, or context.

Range (functions)

The set of all possible outputs of a function.

Quadratic function

A function of the form f(x) = ax^2 + bx + c whose graph is a parabola.

Vertex form

The quadratic form $f(x) = a(x - h)^2 + k$, which shows the vertex directly as $(h, k)$.

Vertex

The turning point of a parabola; it is a minimum if $a > 0$ and a maximum if $a < 0$.

Quadratic formula

A formula for solving $ax^2 + bx + c = 0$: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Discriminant

The expression $b^2 - 4ac$ in the quadratic formula; it determines the number of real solutions.

Difference of squares

A factoring pattern $a^2 - b^2 = (a - b)(a + b)$.

Rational expression

A fraction made of polynomials, such as $\frac{x^2 - 1}{x - 1}$; the original denominator can never equal $0$.

Exponent rules

Key laws such as $a^m \cdot a^n = a^{m+n}$, $\frac{a^m}{a^n} = a^{m-n}$, and $(a^m)^n = a^{mn}$.

Exponential function

A function of the form y = a*b^x, where a is the initial value and b is the growth or decay factor.

Ratio

A comparison of two quantities by division, such as A:B = 3:5 or A/B = 3/5.

Constant of proportionality

The constant k in a proportional relationship y = kx.

Unit rate

A rate expressed per 1 unit, such as dollars per notebook or miles per hour.

Percent change

The change from an original value to a new value, computed as (new - original) / original * 100%.

Reverse percent

A method for finding an original value from a final value after a percent change by dividing by the percent multiplier.

Weighted average

An average that accounts for different weights or amounts, found with (av1 + bv2) / (a + b) or by using percent weights.

Direct variation

A relationship of the form y = kx, where y varies directly with x.

Inverse variation

A relationship of the form y = k/x, where y varies inversely with x.

Mean

The arithmetic average: the sum of the values divided by the number of values.

Median

The middle value in an ordered data set, or the average of the two middle values if there is an even number of data points.

Range (statistics)

A measure of spread found by subtracting the minimum value from the maximum value.

Standard deviation

A measure of how spread out data values are around the mean; more dispersion means a larger standard deviation.

Line of best fit

A line drawn through a scatterplot to model the overall linear trend.

Residual

The difference between an actual value and a predicted value: residual = actual - predicted.

Conditional probability

The probability of event A given event B, computed as P(A | B) = P(A and B) / P(B).

Random sample

A sample chosen so members of the population have a fair chance of selection; it supports generalizing results to the population.

Randomized experiment

An experiment with random assignment to treatments; it supports cause-and-effect conclusions.

Pythagorean theorem

For a right triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.

Similar triangles

Triangles with equal corresponding angles and proportional corresponding side lengths.

Arc length

The length of a part of a circle's circumference; for a central angle $\theta$ in degrees, $\text{arc length} = \left( \frac{\theta}{360} \right) \cdot 2 \pi r$.

Midpoint

The point halfway between $(x_1, y_1)$ and $(x_2, y_2)$: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.

Right-triangle trigonometry

For an acute angle $\theta$, $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$.

Imaginary unit

The number $i$ defined by $i^2 = -1$; powers of $i$ repeat every 4.

Complex number

A number of the form a + bi, where a and b are real numbers.