Integers are all whole numbers, including negatives and zero: ...−3, −2, −1, 0, 1, 2, 3...

Absolute value is the distance from zero on the number line. It is always positive (or zero).

Adding integers: same signs → add the numbers and keep the sign.

Adding integers: different signs → subtract the numbers and keep the sign of the larger absolute value.

When a math problem has multiple operations, you must solve them in this exact order:

Converting:

Adding fractions with different denominators requires finding the Least Common Denominator (LCD).

Ratio: a comparison of two quantities. Can be written as 3:4 or 3/4.

Proportion: a statement that two ratios are equal.   3/4 = 6/8

To solve a proportion, use cross-multiplication:

Solving an equation means finding the value of the variable that makes both sides equal.

The Balance Rule: whatever you do to one side, you must do to the other side.

Think of negative numbers as positions to the left of zero on a number line. The further left, the smaller the number.

Rules for multiplying and dividing:

Exponent: tells you how many times to multiply the base by itself.

Scientific notation is a shorthand for very large or very small numbers. Move the decimal so there is one non-zero digit to the left, then count the moves as the exponent of 10.

One-step equations require a single inverse operation to isolate the variable.

Two-step equations require two operations, working in reverse order of PEMDAS.

Inequalities are solved just like equations, with one critical rule:

Symbols: < (less than)   > (greater than)   ≤ (less than or equal)   ≥ (greater than or equal)

Graphing on a number line: Open circle for < and > (does not include that point). Closed circle for ≤ and ≥ (includes that point).

Slope describes how steep a line is: the rise (vertical change) over the run (horizontal change).

A positive slope goes up from left to right. A negative slope goes down from left to right.

How to graph: Start by plotting the point (0, b) on the y-axis. Then use the slope m = rise/run to find a second point.

To graph a line in slope-intercept form (y = mx + b):

x-intercept: set y = 0 and solve for x.

A system of equations is two equations with two variables. The solution is the point (x, y) where both equations are true at the same time.

Substitution method: Solve one equation for one variable, then substitute that expression into the other equation.

Elimination method: Add or subtract the two equations to cancel out one variable entirely.

A polynomial is an expression made of terms with variables and exponents. Examples: 3x² + 2x − 5

Adding/Subtracting: Combine like terms (same variable and exponent).

Multiplying (FOIL method): For two binomials, multiply First, Outer, Inner, Last terms.

Greatest Common Factor (GCF): Always check if you can factor out a GCF first.

Factoring trinomials (ax² + bx + c where a = 1): Find two numbers that multiply to c and add to b.

Standard form:

The graph of a quadratic is called a parabola. It has a turning point called the vertex. If a > 0, it opens upward; if a < 0, it opens downward.

Solving by factoring is the reverse of FOIL.

When factoring is not obvious, use the quadratic formula to solve any quadratic equation.

The Discriminant is b² − 4ac. It tells you how many real solutions exist:

A function assigns exactly one output to every input. Think of it as a machine: put in a number, get one number out.

Domain: all allowed inputs.   Range: all possible outputs.

Vertical Line Test: a graph represents a function if any vertical line crosses it at most once.

b > 1: exponential growth (the output gets bigger and bigger fast)

0 < b < 1: exponential decay (the output shrinks toward zero)

Real-world examples: compound interest, population growth, radioactive decay.

A logarithm is the inverse of an exponent. It asks: "What power do I raise the base to, to get this number?"

Common log: log₁₀ written as just "log"    Natural log: ln (base e ≈ 2.718)

Key log rules:

Arithmetic sequence: each term is found by adding the same number (common difference d).

Geometric sequence: each term is found by multiplying by the same number (common ratio r).

Sum of an arithmetic series:

Angle types:

Angle pairs:

Transversal crossing parallel lines creates pairs of equal angles:

Types by sides:

Triangle Inequality Theorem: the sum of any two sides must be greater than the third side.

In a right triangle, the square of the hypotenuse (c) equals the sum of the squares of the other two sides.

■ Rectangles and Squares

△ Triangles

◆ Parallelogram and Trapezoid

● Circle

■ Rectangular Prism (Box)

● Cylinder

▲ Cone and Sphere

Translation (slide): Add or subtract from the coordinates.

Reflection (flip):

Rotation:

Dilation (scale): Multiply both coordinates by the scale factor k.

Trigonometry with right triangles requires identifying three sides relative to the angle you are working with (call it θ):

The same triangle has different "opposite" and "adjacent" labels depending on which angle you pick as θ.

The three core trig ratios for a right triangle:

The unit circle is a circle with radius 1 centered at the origin. For any angle θ, the point on the circle is (cos θ, sin θ).

Key angles to memorize:

When graphed, sine and cosine create smooth wave patterns that repeat. This is called being periodic.

Amplitude: the height of the wave from center to peak. For y = A sin(x), amplitude = |A|.

Period: the horizontal length of one complete cycle. For y = sin(Bx), period = 360° / B.

Inverse trig functions let you find an angle when you know the ratio. They undo the trig function.

Angle of elevation: the angle measured upward from horizontal to a line of sight.

Angle of depression: the angle measured downward from horizontal to a line of sight.

The key to word problems is drawing the right triangle and correctly labeling the sides relative to the given angle.