Essential Algebra and Calculus Formulas and Concepts

Coordinate Geometry Formulas

For points A(x₁, y₁) and B(x₂, y₂), the distance formula is:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Midpoint formula: ((x₁ + x₂)/2, (y₁ + y₂)/2)

Functions and Calculus Basics

• Difference quotient: f(x+h) - f(x) / h
• X-intercepts: Not imaginary, written as (x, y).
• Solutions/Roots/Zeros: Can be imaginary, written as x = ___.

Financial and Parent Functions

• Compound interest: A = P(1 + r/n)^nt (r must be a decimal).
• Continuous compound interest: A = Pe^rt
• Parent function y = x²: Domain: all real numbers, Range: y ≥ 0.

• Parent function y = √x: Domain: inside ≥ 0, Range: y ≥ 0.

Transformations and Analysis

Key: For transformations, ensure x inside parentheses is always positive; factor negatives out front.

• h(x) = -3^2(x+4): Left 4, flip over x-axis, horizontal compress by 2.
• f(x) = 3e^x+1: Left 1, vertical stretch by 3.
• g(x) = 2^-x+1: Up 1, flip over y-axis.

Note: f(x) = 2^x (growth) vs 2^-x (decay).

Advanced Concepts

• Local min/max: Write as (x, y).
• Increasing/decreasing intervals: Write in terms of x.
• Inverse functions f^-1(x): Switch x and y, then solve for y. Include the domain of both the function and the inverse.
• Composition: (f ∘ g) = f(g(x)).
• Interval notation: Use (-∞, 2) ∪ (2, ∞). Use brackets [ ] only when the value is an x-intercept and the inequality includes "equal to". Use parentheses for infinity or non-inclusive values.
• Point-slope form: Y = y₁ + m(X - x₁).
• Parallel lines: Same slopes.
• Perpendicular lines: Negative reciprocal slopes.

Quadratic and Complex Solutions

Revenue: R = price × items sold. Find the vertex to determine the ideal price.

Complex numbers: If you have one imaginary root, there must be another. Factor until imaginaries are gone. Powers of i: i¹=i, i²=-1, i³=-i, i⁴=1 (cycle repeats).

Completing the Square (CTS): Move the constant to the right, take half of the x-coefficient, square it, add to both sides, rewrite as a squared term, and solve.

Exponential and Rational Functions

Exponential graph f(x) = a^x (a > 0, a ≠ 1): Domain: (-∞, ∞), Range: (0, ∞), y-intercept: (0, 1), Horizontal Asymptote: y = 0.

Asymptote Rules

1. Hole: Cancel common factors in numerator and denominator, then solve for y.
2. Vertical Asymptote (VA): Set denominator to 0.
3. Horizontal Asymptote (HA):
   • Same degree: Divide leading coefficients.
   • Small/Big: HA is y = 0.
   • Big/Small: No HA (check for slant asymptote via division).