Algebra 1 - Formula Sheet Notes

Properties

• Distributive Property: $a(b+c)=ab+ac$

• Commutative Property: $a+b=b+a$

• Associative Property: $(a+b)+c = a+(b+c) \text{ and } (ab)c = a(bc)$

• Note on line-related content: The transcript mentions the slope and line concepts; see the sections on slope and line forms below for details.

Slope and Line Concepts

• Slope (m) of a line: $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$

• Slope interpretation: rise over run; indicates steepness and direction of a line.

• Slope-intercept form of a line: $y = mx + b$

• Point-slope form of a line: $y - y<em>1 = m(x - x</em>1)$

• (Implicit in the transcript) The line concept also encompasses the idea of a linear function with slope m and intercept b.

Exponents

• Product of powers: $a^m \cdot a^n = a^{m+n}$

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

• Power of a power: $\bigl(a^m\bigr)^n = a^{mn}$

• These rules are key for simplifying expressions with the same base.

Quadratic Formula

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Discriminant: $\Delta = b^2 - 4ac$

  • If \Delta > 0, two real roots.

  • If $\Delta = 0$, one real root (a repeated root).

  • If \Delta < 0, two complex roots.

• Notes:

  • Denominator is $2a$ (with $a \neq 0$ for a quadratic equation).

  • The formula yields all real or complex solutions to $ax^2 + bx + c = 0$.

Arithmetic & Geometric Sequences

• Arithmetic Sequence

  • Definition: consecutive terms differ by a constant amount (the common difference).

  • Formula for the nth term: $a<em>n = a</em>1 + (n-1)d$

  • Here:\n - $a_1$ = first term

  • $d$ = common difference

• Geometric Sequence

  • Definition: consecutive terms are multiplied by a constant factor (the common ratio).

  • Formula for the nth term: $a<em>n = a</em>1 r^{\,n-1}$

  • Here:\n - $a_1$ = first term

  • $r$ = common ratio

• Quick connections:

  • Arithmetic sequence grows linearly with $n$; geometric sequence grows (or decays) exponentially with $n$.

Quick reference of key forms and ideas

• Linear forms: $y = mx + b$; slope $m$ measures steepness, intercept $b$.

• Point-slope form: $y - y<em>1 = m(x - x</em>1)$; useful when a point
  $(x<em>1, y</em>1)$ on the line is given.

• Quadratic standard setup: $ax^2 + bx + c = 0$ with discriminant $\Delta = b^2 - 4ac$ and solution via the Quadratic Formula.

• Exponent rules summarize how to combine and simplify expressions with the same base $a$ across multiplication, division, and power operations.

• Sequence types (arithmetic vs geometric) determine how terms progress as index $n$ increases:

  • Arithmetic: additive difference $d$

  • Geometric: multiplicative ratio $r$

Examples (illustrative, to reinforce the concepts above)

• Distributive Property:

  • Example: $3\bigl( x + 5 \bigr) = 3x + 15$

• Slope calculation:

  • Points (2,3) and (5,11): $m = \frac{11-3}{5-2} = \frac{8}{3}$

• Quadratic Formula example:

  • Solve $2x^2 + 3x - 2 = 0$:

  • $\Delta = 3^2 - 4\cdot 2\cdot (-2) = 9 + 16 = 25$

  • $x = \frac{-3 \pm \sqrt{25}}{2\cdot 2} = \frac{-3 \pm 5}{4}$

  • Solutions: $x = \frac{1}{2}$ and $x = -2$

• Arithmetic sequence example:

  • If $a<em>1 = 4$ and $d = 3$, then $a</em>n = 4 + (n-1)\cdot 3$

• Geometric sequence example:

  • If $a<em>1 = 7$ and $r = 2$, then $a</em>n = 7\cdot 2^{\,n-1}$

Connections and practical implications

• These formulas form the core toolkit for solving algebra problems involving lines, quadratics, and sequences.

• Understanding the properties (distributive, commutative, associative) enables algebraic manipulation essential for simplifying expressions and solving equations.

• The quadratic formula and discriminant guide the nature and number of roots, informing problem-solving strategies in geometry, physics, and economics.

• Recognizing when a problem involves a linear model (slope-intercept form) versus a quadratic model (quadratic formula) helps choose the appropriate method quickly.