College Algebra Comprehensive Study Guide: Key Concepts and Formulas

{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":2},"content":[{"type":"text","text":"Module R: Review of Basic Algebra"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"R.1 Sets and Real Numbers"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This section introduces the foundational sets of numbers and their properties, which are essential for understanding algebraic operations and classifications."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Natural Numbers (N):"},{"type":"text","text":" {1, 2, 3, ...}"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Whole Numbers:"},{"type":"text","text":" {0, 1, 2, ...}"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Integers (Z):"},{"type":"text","text":" {..., -2, -1, 0, 1, 2, ...}"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Rational Numbers (Q):"},{"type":"text","text":" Numbers expressible as "},{"type":"inlineMath","attrs":{"latex":"\\frac{p}{q}"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"q \\neq 0"}},{"type":"text","text":"; decimals terminate or repeat."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Irrational Numbers:"},{"type":"text","text":" Decimals never terminate or repeat (e.g., "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{5}"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\pi"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Real Numbers:"},{"type":"text","text":" The union of rational and irrational numbers."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Set Notation:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"2 \\in S"}},{"type":"text","text":" means 2 is an element of S; "},{"type":"inlineMath","attrs":{"latex":"A \\subset B"}},{"type":"text","text":" means every element of A is in B."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Roster Notation:"},{"type":"text","text":" Lists elements, e.g., {1, 2, 3}."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Set-Builder Notation:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\{x \\mid x > 5\\}"}},{"type":"text","text":" means all x such that x > 5."}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Example:"},{"type":"text","text":" 5, "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{4} = 2"}},{"type":"text","text":", 0, and "},{"type":"inlineMath","attrs":{"latex":"\\sqrt[3]{216} = 6"}},{"type":"text","text":" are whole numbers."}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"R.2 Interval Notation"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Interval notation is used to describe sets of real numbers, especially for domains and solution sets."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Open Interval:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"(a, b)"}},{"type":"text","text":" means "},{"type":"inlineMath","attrs":{"latex":"a < x < b"}},{"type":"text","text":" (endpoints excluded)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Closed Interval:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"[a, b]"}},{"type":"text","text":" means "},{"type":"inlineMath","attrs":{"latex":"a \\leq x \\leq b"}},{"type":"text","text":" (endpoints included)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Half-Open:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"[a, b)"}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":"(a, b]"}},{"type":"text","text":" includes one endpoint."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Infinity:"},{"type":"text","text":" Always use parentheses: "},{"type":"inlineMath","attrs":{"latex":"(a, \\infty)"}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":"(-\\infty, b)"}},{"type":"text","text":"."}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Example:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"-5 < x < 5 \\rightarrow (-5, 5)"}},{"type":"text","text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"R.3 Properties of Real Numbers and Absolute Value"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Understanding these properties is crucial for manipulating algebraic expressions."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Commutative:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"a + b = b + a"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"ab = ba"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Associative:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"a + (b + c) = (a + b) + c"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Identity:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"a + 0 = a"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"a \\cdot 1 = a"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Inverse:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"a + (-a) = 0"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"a \\cdot \\frac{1}{a} = 1"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Distributive:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"a(b + c) = ab + ac"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Absolute Value:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"|a|"}},{"type":"text","text":" is the distance from 0; "},{"type":"inlineMath","attrs":{"latex":"|a - b|"}},{"type":"text","text":" is the distance between a and b."}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Example:"},{"type":"text","text":" Distance between -7 and 4: "},{"type":"inlineMath","attrs":{"latex":"|-7 - 4| = 11"}},{"type":"text","text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"R.4 Integer Exponents"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Exponent rules are fundamental for simplifying expressions and solving equations."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"x^0 = 1"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"x^{-n} = \\frac{1}{x^n}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"x^m \\cdot x^n = x^{m+n}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\frac{x^m}{x^n} = x^{m-n}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"(x^m)^n = x^{mn}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"(xy)^m = x^m y^m"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\left(\\frac{x}{y}\\right)^m = \\frac{x^m}{y^m}"}}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Example:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"2^{-5} = \\frac{1}{32}"}}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"R.5 Scientific Notation and Order of Operations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Scientific notation expresses very large or small numbers compactly. Order of operations ensures correct calculation."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Scientific Notation:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"N \\times 10^m"}},{"type":"text","text":" with "},{"type":"inlineMath","attrs":{"latex":"1 \\leq N < 10"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Order of Operations:"},{"type":"text","text":" Parentheses → Exponents → Multiplication/Division → Addition/Subtraction"}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Example:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"0.0000000068 = 6.8 \\times 10^{-9}"}}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"R.6 Polynomials"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Polynomials are algebraic expressions with terms of non-negative integer exponents."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Degree:"},{"type":"text","text":" Highest sum of exponents in any term."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Types:"},{"type":"text","text":" Monomial (1 term), Binomial (2 terms), Trinomial (3 terms)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Operations:"},{"type":"text","text":" Combine like terms, distribute, FOIL for binomials."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Special Products:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"(A + B)^2 = A^2 + 2AB + B^2"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"(A - B)^2 = A^2 - 2AB + B^2"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"(A + B)(A - B) = A^2 - B^2"}}]}]}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Example:"},{"type":"text","text":" Degree of "},{"type":"inlineMath","attrs":{"latex":"6w^9 + 10w^{10}p^4 + 3w^3p^3 + 4w^3 - 4"}},{"type":"text","text":" is 14."}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"R.7 Factoring"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factoring rewrites polynomials as products of simpler expressions."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"GCF:"},{"type":"text","text":" Factor out the greatest common factor first."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Grouping:"},{"type":"text","text":" For four terms, group and factor."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Trinomials:"},{"type":"text","text":" Find two numbers that multiply to c and add to b."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"ac Method:"},{"type":"text","text":" For "},{"type":"inlineMath","attrs":{"latex":"ax^2 + bx + c"}},{"type":"text","text":", split the middle term."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Special Patterns:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"A^2 - B^2 = (A + B)(A - B)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"A^2 \\pm 2AB + B^2 = (A \\pm B)^2"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"A^3 + B^3 = (A + B)(A^2 - AB + B^2)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"A^3 - B^3 = (A - B)(A^2 + AB + B^2)"}}]}]}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Example:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"s^2 - 10s + 21 = (s - 3)(s - 7)"}}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"R.8 Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solving equations involves isolating the variable using algebraic properties."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Linear:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"ax + b = 0"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Zero-Product Property:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"ab = 0"}},{"type":"text","text":" implies "},{"type":"inlineMath","attrs":{"latex":"a = 0"}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":"b = 0"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Square-Root Principle:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"x^2 = k"}},{"type":"text","text":" implies "},{"type":"inlineMath","attrs":{"latex":"x = \\pm \\sqrt{k}"}}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Example:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"8y - 7 = 23 - 2y \\rightarrow y = 3"}}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"R.9 Rational Expressions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rational expressions are quotients of polynomials. Simplification and operations require factoring and finding common denominators."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Domain:"},{"type":"text","text":" Exclude values making the denominator zero."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Simplify:"},{"type":"text","text":" Factor numerator and denominator, cancel common factors."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Operations:"},{"type":"text","text":" Multiply/divide by factoring; add/subtract using LCD."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Complex Fractions:"},{"type":"text","text":" Multiply numerator and denominator by LCD of all denominators."}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Example:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\frac{p^2 - 49}{p^2 - 14p + 49} = \\frac{p + 7}{p - 7},\\ p \\neq 7"}}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"R.10 Radicals and Rational Exponents"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Radicals and rational exponents are alternative ways to express roots and powers."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sqrt[n]{ab} = \\sqrt[n]{a} \\cdot \\sqrt[n]{b}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sqrt[n]{a/b} = \\frac{\\sqrt[n]{a}}{\\sqrt[n]{b}}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sqrt[n]{x^m} = x^{m/n}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Even index: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt[n]{x^n} = |x|"}},{"type":"text","text":"; Odd index: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt[n]{x^n} = x"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rationalize denominators by multiplying by the conjugate if necessary."}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Example:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{6} \\cdot \\sqrt{14} = 2\\sqrt{21}"}}]},{"type":"heading","attrs":{"textAlign":null,"level":2},"content":[{"type":"text","text":"Module 1: Graphs, Functions, and Models"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"1.1 Introduction to Graphing"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Graphing is a visual representation of equations and functions in the coordinate plane."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Quadrants:"},{"type":"text","text":" I (+,+), II (-,+), III (-,-), IV (+,-)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Intercepts:"},{"type":"text","text":" x-intercept: set y = 0; y-intercept: set x = 0"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Distance Formula:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Midpoint Formula:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\left(\\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Circle Equation:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"(x - h)^2 + (y - k)^2 = r^2"}}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Example:"},{"type":"text","text":" Distance from (−2, 2) to (3, −6): "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{89} \\approx 9.4"}}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"1.2 Functions and Graphs"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"A function assigns each input exactly one output. The graph of a function passes the vertical-line test."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Function Notation:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"f(x)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Domain:"},{"type":"text","text":" All real numbers except those making denominators zero or even roots negative."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Vertical-Line Test:"},{"type":"text","text":" If any vertical line crosses the graph more than once, it is not a function."}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Example:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"f(x) = 2x^2 - 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