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Understanding the number system — integers, rational numbers, and how they relate to each other on the number line.
Absolute value is the distance a number is from zero on the number line — it is always positive or zero.
Adding numbers follows three scenarios: both positive (add, keep positive), both negative (add, keep negative), or different signs (subtract the smaller absolute value and keep the sign of the larger).
To subtract any number, use Keep · Change · Change (KCC): keep the first number, change subtraction to addition, change the sign of the second number. Then follow the addition rules.
When multiplying two numbers: same signs give a positive result, different signs give a negative result.
Division follows the same sign rules as multiplication: same signs give a positive quotient, different signs give a negative quotient. Divide the absolute values, then apply the sign rule.
Exponents tell you how many times to multiply a base by itself. Fractional exponents connect to roots: x^(1/2) = √x, x^(1/3) = ∛x. The numerator is the power, the denominator is the root.
A radical (√) asks: what number multiplied by itself gives this value? Simplifying radicals means pulling out perfect square factors.
Monomials can only be added if they are "like terms" — same variable and same exponent. Add the coefficients and keep the variable part.
To multiply monomials: multiply the coefficients together and add the exponents of matching variables.
To divide monomials: divide the coefficients and subtract the exponents of matching variables.
A monomial is a single term with a coefficient and variables raised to whole-number exponents. The Power of a Power Rule states: (xᵃ)ᵇ = xᵃˣᵇ — when raising a power to another power, multiply the exponents.
A one-step equation is solved by performing a single operation — addition, subtraction, multiplication, or division — to isolate the variable.
Two-step equations require two operations to isolate the variable. Always undo addition/subtraction first, then multiplication/division.
Multi-step equations may require combining like terms, using the distributive property, and moving variables to one side before solving.
Unit conversion uses multiplication by conversion factors — fractions equal to 1 — to change units without changing the value. To convert mph to m/s: multiply by 1609.34 meters/mile and divide by 3600 seconds/hour.
A ratio compares two quantities. A proportion says two ratios are equal. Solve proportions using cross-multiplication: if a/b = c/d, then ad = bc.
Percent means "per hundred." To convert: percent ÷ 100 = decimal; decimal × 100 = percent; fraction → divide → decimal → multiply by 100.
Graphing an inequality on a number line shows all solutions. Use an open circle for > or < (not equal), and a closed circle for ≥ or ≤ (equal included). Shade in the direction of the solutions.
Solve multi-step inequalities like equations — but remember: when you multiply or divide both sides by a negative number, FLIP the inequality sign!
The coordinate plane has a horizontal x-axis and vertical y-axis. Points are written as (x, y) — x tells you how far left/right, y tells you how far up/down.
Slope measures the steepness of a line. It equals rise over run: m = (y₂ − y₁) / (x₂ − x₁). Positive slope goes up left to right; negative slope goes down.
Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals of each other (flip and negate).
Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept (where the line crosses the y-axis).
Graph a linear inequality by first graphing the boundary line (solid for ≤ or ≥, dashed for < or >), then shading the region that satisfies the inequality.
The graph of y = x² is a U-shaped curve called a parabola. It is symmetric about the y-axis with its vertex at the origin (0, 0).
Standard form: y = ax² + bx + c. The axis of symmetry is x = −b/(2a). The vertex x-coordinate is found with this formula, then plug back in to find y.
Vertex form is y = a(x − h)² + k, where (h, k) is the vertex. This form makes it easy to identify the vertex and graph the parabola.
End behavior describes what happens to a polynomial as x → +∞ and x → −∞. It depends on the leading coefficient and the degree (even or odd).
The distributive property lets you multiply a term by everything inside parentheses: a(b + c) = ab + ac. With polynomials, multiply each term carefully.
The height of an object thrown upward is modeled by h(t) = −16t² + v₀t + h₀ (in feet), where v₀ is initial velocity and h₀ is initial height. Solve to find height at a given time, when it hits the ground, or when it reaches max height.
The standard equation of a circle is (x − h)² + (y − k)² = r², where (h, k) is the center and r is the radius.
Perimeter is the total distance around a polygon. Circumference is the distance around a circle: C = 2πr or C = πd.
Area measures the space inside a 2D shape. Key formulas: Rectangle = lw, Triangle = ½bh, Circle = πr², Trapezoid = ½(b₁+b₂)h.
Volume measures the space inside a 3D shape. Key formulas: Rectangular prism = lwh, Cylinder = πr²h, Cone = ⅓πr²h, Sphere = (4/3)πr³.
The three interior angles of any triangle always add up to 180°. Use this to find a missing angle when you know the other two.
Similar triangles have the same shape but different sizes. Their corresponding sides are proportional. Set up a proportion with corresponding sides to find missing lengths.
Two triangles are congruent if they have the same size and shape. Congruence postulates: SSS (3 sides), SAS (2 sides + included angle), ASA (2 angles + included side), AAS (2 angles + non-included side), HL (right triangles: hypotenuse + leg).
When a transversal crosses two parallel lines, it creates 8 angles. Alternate interior angles are equal, corresponding angles are equal, co-interior (same-side interior) angles are supplementary (add to 180°).
In a right triangle, SOH-CAH-TOA: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. Use one triangle to find all three ratios.
Trigonometry solves real-world problems involving angles of elevation or depression. Set up a right triangle, identify the known angle and side, then use sin, cos, or tan to find the unknown.
A translation slides every point of a figure the same distance in the same direction. It is described as (x + a, y + b) — add a to every x-coordinate and b to every y-coordinate.
A reflection flips a figure over a line of reflection. Over the x-axis: (x, y) → (x, −y). Over the y-axis: (x, y) → (−x, y). Over y = x: (x, y) → (y, x).
A rotation turns a figure around a center point. Common rotations about the origin: 90° clockwise: (x,y)→(y,−x). 90° counterclockwise: (x,y)→(−y,x). 180°: (x,y)→(−x,−y).
An arithmetic sequence has a constant difference (d) between terms. The nth term formula is: aₙ = a₁ + (n − 1)d, where a₁ is the first term.
A geometric sequence has a constant ratio (r) between terms. The nth term formula is: aₙ = a₁ × r^(n−1).
Mean = average (sum ÷ count). Median = middle value when sorted. Mode = most frequent value. Range = largest − smallest.
Probability = (number of favorable outcomes) / (total number of outcomes). It is always between 0 and 1. A spinner with equal sections makes a great model.
Two events are independent if one does not affect the other. P(A and B) = P(A) × P(B). Example: flipping a coin and rolling a die are independent.
Events are dependent when the outcome of the first affects the second. P(A then B) = P(A) × P(B|A), where P(B|A) means probability of B given A already happened.
Two-way tables organize data by two categories. The KEY skill is knowing what goes on the bottom of your fraction — and that depends on the type of question: • P(event) — no "given" — bottom is the GRAND TOTAL (whole survey). • P(A AND B) — both things happen — bottom is the GRAND TOTAL. • P(A | B) — "given B" — bottom is the TOTAL FOR B ONLY. You zoom in to just that row or column and ignore everyone else. The word GIVEN tells you to shrink your world. Instead of the whole survey, you only look at the group that was given to you.
