Algebra I Study Notes

Laws of Exponents

• Anything raised to the power of zero equals 1: $x^0 = 1$
• When multiplying, multiply coefficients and add exponents of the same base: $2x^3 imes 4x^2 = 8x^5$
• When raising to a power, distribute and multiply exponents: $(3x^4)^2 = 9x^8$
• When dividing, divide coefficients and subtract exponents: $\frac{6x^4}{2x^3} = 3x^{1}$
• A negative exponent means to flip the base: $x^{-2} = \frac{1}{x^2}$
• A fractional exponent indicates a root: $x^{\frac{1}{2}} = \sqrt{x}$

Factoring Trinomials

• For a trinomial of the form $ax^2 + bx + c$, follow these steps:
  1. Factor out GCF if applicable.
  2. Identify coefficients: $a, b, c$.
  3. Calculate $ac$ to find factors that add to $b$.
  4. Rewrite as 4 terms and factor by grouping.
  5. Example: $12x^2 - 10x + 2 = 2(6x^2 - 5x + 1) = 2(2x-1)(3x-1)$

Graphing Quadratics

• Vertex: Point at which the function changes direction.
• Axis of Symmetry: Vertical line through the vertex.
• X-Intercepts: Points where the graph crosses the x-axis.
• Example Vertex: (1, 4), X-Intercept: (3, 0)

Polynomial Operations

• For addition and subtraction, combine like terms:
  $(3x^2 - 5x + 7) + (x^2 + 2x - 1) = 4x^2 - 3x + 6$
• For multiplication, use distribution:
  $(x+2)(x-3) = x^2 - x - 6$

Exponential Functions

• General form: $y = ab^x$
• Characteristics:
  • $a$ is the y-intercept, $b$ is the base.
  • Asymptote: line the function approaches but never touches (e.g., $y=0$).

Quadratic Solutions

• Methods: Graphing, Factoring, Completing the Square, Quadratic Formula:
  $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$
• Effects on Graph:
  • Vertical Stretch/Compression by adjusting $a$.
  • Horizontal Stretch/Compression by adjusting $b$.
  • Shifts by adjustments with $c$ (right/left) and $d$ (up/down).

Domain & Range

• Domain: all possible $x$ values; Range: all possible $y$ values.
• Continuous functions have unbroken graphs.
• For discrete, specific values are used (e.g., $ext{Domain: } {-6, -5, -4, \ldots, 6}$).

Finding Slope

• Slope formula: $m = \frac{\text{rise}}{\text{run}}$
• Forms of linear equations:
  • Slope-intercept: $y = mx + b$
  • Point-slope: $y - y<em>1 = m(x - x</em>1)$
  • Standard: $Ax + By = C$

Solving Equations & Inequalities

• Distributing and combining terms with variables on both sides:
  E.g., $-13x - 53 = -2(2x - 14)$.
• Flip inequality symbol when multiplying/dividing by a negative: e.g., -5(2x - 3) > 8x - 21

Systems of Linear Equations

• Solve by elimination or substitution.
• Infinite solutions: lines overlap (e.g., same equation).
• No solution: lines are parallel (e.g. inconsistent equations).

Additional Notes

• A relation is a function if each $x$-value is unique (Vertical Line Test).
• Key features of linear functions include slope and intercepts.
• Solutions to linear inequalities are represented in shaded graphs based on inequality types.