Essential Algebra Formulas and Graphing Techniques

Posted on May 31, 2026 in Mathematics

Algebraic Formulas and Equations

Quadratic and Polynomial Functions

  • AOS: x = -b/2a
  • Quadratic Formula: x = (-b ± √b² – 4ac) / 2a
  • Discriminant (b² – 4ac):
    • > 0: Two solutions
    • < 0: No real solutions
    • = 0: One solution
  • Completing the Square: Take 1/2 of b, square it, and add to both sides. Rewrite as (x ± 1/2b)².

Sequences and Financial Math

  • Arithmetic Sequence: aₙ = a₁ + d(n – 1)
  • Geometric Sequence: aₙ = a₁ * rⁿ⁻¹
  • Compound Interest (Annually): P(1 + r)ᵗ
  • Compound Interest (Quarterly): P(1 + r/n)ⁿᵗ
  • Continuous Interest: Peʳᵗ
  • Note: Convert all percentages to decimals (e.g., 1.8% = 0.018).

Systems and Equations

  • Elimination: Align equations to cancel one variable.
  • Substitution: Replace one variable with its equivalent expression.
  • Absolute Value Equations: Isolate the brackets, then set the expression equal to both the positive and negative values of the other side.
  • Inequalities: Flip the inequality sign when multiplying or dividing by a negative number.
  • Compound Inequalities: Solve for x in both equations and graph the solution on a number line.

Radicals and Exponents

  • Negative Exponents: Take the reciprocal of the base and make the exponent positive.
  • Negative/Fractional Exponents: Reciprocate if negative. The numerator is the power; the denominator is the root.
  • Multiplying/Dividing Radicals: Multiply/divide outside numbers and inside values separately, then simplify.
  • Rationalizing Denominators: Multiply the top and bottom by the radical or the conjugate (flip the sign of the second term).
  • Radical Equations: Always square both sides completely and check for extraneous solutions.

Functions and Notation

  • Interval Notation: Use brackets for inclusive values (≥, ≤) and parentheses for exclusive values (<, >). Use U for unions.
  • Fog Notation: (f ∘ g)(2) = f(g(2)). Solve the inner function first.

Graphing Techniques

Linear and Piecewise Functions

  • Ax + By = C: Solve for x-intercept (set y=0) and y-intercept (set x=0).
  • Vertical Lines: x = number (undefined slope).
  • Horizontal Lines: y = number (zero slope).
  • Point-Slope Form: Use y – y₁ = m(x – x₁) or solve for b in y = mx + b.
  • Piecewise Functions: Draw dotted lines at boundary values and graph segments accordingly.

Parabolas

  • Upward Opening: Occurs when the leading coefficient is positive.
  • Vertex Form: y = a(x – h)² + k (flip the sign for h).
  • Roots: Factor the equation; roots are the x-intercepts.
  • Graphing: Find the AOS, calculate the vertex, and create a table for additional points.