Comprehensive Secondary Mathematics Study Guide: Order of Operations, Exponents, Functions, and Relations

Order of Operations, Expression Simplification, and Linear Equations

• Order of Operations (PEMDAS/BODMAS)

  • Numerical expressions must be evaluated in a strict hierarchical order:

  • Parentheses and Brackets: Evaluate terms within innermost grouping symbols first.

  • Exponents: Compute powers and roots.

  • Multiplication and Division: Perform operations from left to right in the order they appear.

  • Addition and Subtraction: Perform operations from left to right in the order they appear.

• Arithmetic Evaluation Examples

  • Expression 1: $(45 - 3) \div 6 - ((2 + 1) \times 3) \div (10 - 1)$

  • Inner grouping evaluations:

    • $(45 - 3) = 42$

    • $(2 + 1) = 3 \implies (3 \times 3) = 9$

    • $(10 - 1) = 9$

  • Intermediate expression: $42 \div 6 - 9 \div 9$

  • Division operations: $42 \div 6 = 7$ and $9 \div 9 = 1$

  • Final result: $7 - 1 = 6$

  • Expression 2: $11 - 12 \div (1 + 12 - 4 + 5 - 4 - 4)$

  • Grouping evaluation: $1 + 12 - 4 + 5 - 4 - 4 = 6$

  • Division operation: $12 \div 6 = 2$

  • Final result: $11 - 2 = 9$

  • Expression 3: $5 \div (4 \times 8 - (15 + 13 - 11) - (5 + 5))$

  • Inner grouping evaluations:

    • $15 + 13 - 11 = 17$

    • $5 + 5 = 10$

  • Bracket evaluation: $4 \times 8 - 17 - 10 = 32 - 17 - 10 = 5$

  • Division operation: $5 \div 5$

  • Final result: $1$

  • Expression 4: $(6 \times 2) \div 12 \times (9 - 44 \div 11)(15 - 14)$

  • Grouping evaluations:

    • $(6 \times 2) = 12$

    • $44 \div 11 = 4 \implies (9 - 4) = 5$

    • $(15 - 14) = 1$

  • Left-to-right operations: $12 \div 12 \times 5 \times 1 = 1 \times 5 \times 1$

  • Final result: $5$

• Simplifying Algebraic Expressions & Combining Like Terms

  • Expression 5: $-14 + 8x + 1 + 10x$

  • Combine variable terms: $8x + 10x = 18x$

  • Combine constant terms: $-14 + 1 = -13$

  • Simplified result: $-13 + 18x$

  • Expression 6: $8 - 9 + 5x + 15x$

  • Combine variable terms: $5x + 15x = 20x$

  • Combine constant terms: $8 - 9 = -1$

  • Simplified result: $-1 + 20x$

  • Expression 7: $23n - 5 + 9 + 7n$

  • Combine variable terms: $23n + 7n = 30n$

  • Combine constant terms: $-5 + 9 = 4$

  • Simplified result: $30n - 4$

  • Expression 8: $3v - 7 - 7 - 80v$

  • Combine variable terms: $3v - 80v = -77v$

  • Combine constant terms: $-7 - 7 = -14$

  • Simplified result: $-14 - 77v$

  • Expression 9: $5(6k - 10)$

  • Apply the distributive property: $5 \times 6k - 5 \times 10$

  • Simplified result: $30k - 50$

  • Expression 10: $-3(3p + 8)$

  • Apply the distributive property: $-3 \times 3p + (-3) \times 8$

  • Simplified result: $-9p - 24$

• Solving Single-Variable Linear Equations

  • Equation 11: $-61 = 7n + 2$

  • Subtract $2$ from both sides: $-63 = 7n$

  • Divide both sides by $7$: $n = -9$

  • Solution set: $\{-9\}$

  • Equation 12: $\frac{n}{7} = -15$

  • Multiply both sides by $7$: $n = -15 \times 7$

  • Solution set: $\{-105\}$

  • Equation 13: $8n + 3 = 27$

  • Subtract $3$ from both sides: $8n = 24$

  • Divide both sides by $8$: $n = 3$

  • Solution set: $\{3\}$

  • Equation 14: $-5a + 7 = -28$

  • Subtract $7$ from both sides: $-5a = -35$

  • Divide both sides by $-5$: $a = 7$

  • Solution set: $\{7\}$

Exponent Rules and Algebraic Simplification

• Fundamental Laws of Exponents

  • Product Rule: $a^m \times a^n = a^{m+n}$

  • Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

  • Power of a Power Rule: $(a^m)^n = a^{m \times n}$

  • Power of a Product Rule: $(ab)^n = a^n b^n$

  • Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$ and $\frac{1}{a^{-n}} = a^n$

  • Zero Exponent Rule: $a^0 = 1$ (where $a \neq 0$)

• Simplification Examples with Positive Exponents

  • Problem 1: $3x^6 \times 2x \times x^6$

  • Multiply coefficients: $3 \times 2 = 6$

  • Add exponents: $x^{6 + 1 + 6} = x^{13}$

  • Result: $6x^{13}$

  • Problem 2: $4x^6 \times 2x \times 3x^{-2}$

  • Multiply coefficients: $4 \times 2 \times 3 = 24$

  • Add exponents: $x^{6 + 1 + (-2)} = x^5$

  • Result: $24x^5$

  • Problem 3: $\frac{4p^{10}}{7p^{-7}}$

  • Move negative exponent to numerator: $p^{10 - (-7)} = p^{17}$

  • Result: $\frac{4p^{17}}{7}$

  • Problem 4: $\frac{10n^7}{4n^6}$

  • Simplify fractional coefficient: $\frac{10}{4} = \frac{5}{2}$

  • Subtract exponents: $n^{7 - 6} = n^1$

  • Result: $\frac{5n}{2}$

  • Problem 5: $\frac{n^{-10}}{n^4}$

  • Apply quotient rule: $n^{-10 - 4} = n^{-14}$

  • Convert to positive exponent: $\frac{1}{n^{14}}$

  • Problem 6: $\frac{x^{-2}}{9x^{10}}$

  • Apply exponent rule: $x^{-2 - 10} = x^{-12}$

  • Result: $\frac{1}{9x^{12}}$

  • Problem 7: $(3b^2)^4$

  • Distribute exponent to coefficient and variable: $3^4 \times (b^2)^4$

  • Compute power: $81b^{2 \times 4} = 81b^8$

  • Problem 8: $(2n^8)^6$

  • Distribute exponent: $2^6 \times (n^8)^6$

  • Compute power: $64n^{8 \times 6} = 64n^{48}$

  • Problem 9: $\frac{8x^4 \times 9x^{10}}{8x^2}$

  • Simplify numerator: $72x^{14}$

  • Divide by denominator: $\frac{72x^{14}}{8x^2} = 9x^{14 - 2} = 9x^{12}$

  • Problem 10: $\frac{8r^9}{7r^{-6} \times 4r^{-3}}$

  • Simplify denominator: $7 \times 4 \times r^{-6 + (-3)} = 28r^{-9}$

  • Evaluate expression: $\frac{8r^9}{28r^{-9}} = \frac{2}{7} r^{9 - (-9)} = \frac{2r^{18}}{7}$

  • Problem 11: $4x(10 - 8x)$

  • Apply distributive law: $4x \times 10 - 4x \times 8x$

  • Result: $40x - 32x^2$

Parent Function Families and Function Evaluation

• Parent Function Family Identification

  • Quadratic Function Family (quad):

  • Characterized by a parabolic curve ($y = ax^2 + bx + c$).

  • Example: Parabola opening downward with vertex at $(2.5, 5.25)$.

• Linear Function Family (linear):

  • Characterized by a continuous straight line ($y = mx + b$).

  • Example: Line passing through points $(0, -3)$ and $(6, -1)$.

• Exponential Function Family (exponential):

  • Characterized by a continuous curve featuring a horizontal asymptote ($y = a b^x + c$).

  • Example: Curve approaching horizontal asymptote at $y = 6$ on the left and decreasing towards negative infinity on the right.

• Absolute Value Function Family (abs value):

  • Characterized by a distinctive V-shaped curve ($y = a|x - h| + k$).

  • Example: Inverted V-shape with a vertex at $(2, -1)$.

• Evaluating Functions for Given Input Values

  • Function 5: $f(x) = -4x + 5$

  • Evaluate $f(3)$: $f(3) = -4(3) + 5 = -12 + 5 = -7$

  • Function 6: $f(x) = x^2 - 2x - 4$

  • Evaluate $f(1)$: $f(1) = (1)^2 - 2(1) - 4 = 1 - 2 - 4 = -5$

  • Function 7: $f(x) = x^2 + 10x + 18$

  • Evaluate $f(-3)$: $f(-3) = (-3)^2 + 10(-3) + 18 = 9 - 30 + 18 = -3$

  • Function 8: $f(x) = -3|x - 2| + 1$

  • Evaluate $f(2)$: $f(2) = -3|2 - 2| + 1 = -3(0) + 1 = 1$

• Review Problems

  • Problem 9: Evaluate $-7 - (21 + 25 - (-27 - 6) - 27)$

  • Innermost evaluation: $-27 - 6 = -33$

  • Parentheses evaluation: $21 + 25 - (-33) - 27 = 21 + 25 + 33 - 27 = 52$

  • Final evaluation: $-7 - 52 = -59$

  • Problem 10: Simplify $26 + 31x + 32x + 26$

  • Group like terms: $(31x + 32x) + (26 + 26) = 63x + 52$

  • Problem 11: Simplify $3(1 - 7n) - 9$

  • Distribute: $3 - 21n - 9 = -6 - 21n$

  • Problem 12: Solve $8n + 6 = 54$

  • Subtract $6$: $8n = 48 \implies n = 6$

  • Problem 13: Solve $1 + 5v = 96$

  • Subtract $1$: $5v = 95 \implies v = 19$

Relations, Functions, Domain, and Range

• Core Definitions

  • Relation: Any set of ordered pairs $(x, y)$ relating inputs to outputs.

  • Function: A specific relation where every element in the domain (input $x$) maps to exactly one element in the range (output $y$).

  • Domain: The set of all possible input values ($x$--coordinates).

  • Range: The set of all possible output values ($y$--coordinates).

  • Vertical Line Test (VLT): A visual criterion stating that a graph represents a function if and only if no vertical line intersects the graph at more than one point.

• Determining Domain and Range from Graphs

  • Graph 1:

  • Continuous ray starting at closed point $(-6, -1)$ extending infinitely to the right and upward.

  • Domain: $[-6, \infty)$

  • Range: $[-1, \infty)$

• Graph 2:

  • Continuous curved function starting at closed point $(-5, -3)$ with absolute minimum at $(0, -4)$ and extending infinitely upward to the right.

  • Domain: $[-5, \infty)$

  • Range: $[-4, \infty)$

• Graph 3:

  • Piecewise line segments from open endpoint $(1, -6)$ down to closed vertex at $(-1, -2)$ and up to open endpoint $(3, 6)$.

  • Domain: $[-1, 3)$

  • Range: $(-6, 6)$

• Graph 4:

  • Closed ellipse centered at $(-2, -1)$.

  • Domain: $[-6, 2]$

  • Range: $[-4, 2]$

• Determining Function Status from Graphs

  • Graph 5:

  • Downward-opening parabola.

  • Application of VLT: Every vertical line crosses the graph at most once.

  • Determination: The relation is a function.

• Graph 6:

  • Angle curve opening horizontally to the right with vertex at $(0, -3)$.

  • Application of VLT: A vertical line (e.g., at $x = 2$) intersects the graph at two distinct points.

  • Determination: The relation is not a function.

• Analyzing Discrete Representations (Tables, Sets, Mapping Diagrams)

  • Representation 7 (Table):

  • Data pairs: $(-6, -2)$, $(-1, -1)$, $(3, 3)$, $(6, -7)$, $(6, -4)$

  • Domain: $\{-6, -1, 3, 6\}$

  • Range: $\{-7, -4, -2, -1, 3\}$

  • Function Status: Not a function (input $6$ maps to two distinct outputs: $-7$ and $-4$).

• Representation 8 (Set of Ordered Pairs):

  • Set: $\{(-4, 3), (-4, -4), (-4, 0), (4, 0), (6, 7)\}$

  • Domain: $\{-4, 4, 6\}$

  • Range: $\{-4, 0, 3, 7\}$

  • Function Status: Not a function (input $-4$ maps to three distinct outputs: $-4$, $0$, and $3$).

• Representation 9 (Set of Ordered Pairs):

  • Set: $\{(-1, 2), (1, 2), (2, 2), (6, 3), (7, -3)\}$

  • Domain: $\{-1, 1, 2, 6, 7\}$

  • Range: $\{-3, 2, 3\}$

  • Function Status: The relation is a function (each input maps to exactly one output value).

• Representation 10 (Mapping Diagram):

  • Mapping details:

    • Input $-7 \rightarrow -3$

    • Input $-1 \rightarrow 7$

    • Input $3 \rightarrow -2, -1, 3$

  • Domain: $\{-7, -1, 3\}$

  • Range: $\{-3, -2, -1, 3, 7\}$

  • Function Status: Not a function (input $3$ points to three different outputs).