Mathematics Cheat Sheet: Algebra, Functions, and Geometry

This cheat sheet covers essential concepts in algebra, functions, and geometry, including exponential growth, logarithmic equations, trigonometric identities, and geometric calculations. It provides quick references for formulas, problem-solving steps, and key definitions.

Core Principles

Exponential growth and decay models.
Logarithm properties and equation solving.
Trigonometric functions and identities.
Geometric formulas for arc length and angles.
Function transformations and graphing.

Action Steps

Identify the type of function or equation.
Apply relevant formulas or properties.
Isolate the variable or solve for the unknown.
Check for extraneous solutions, especially with roots or logarithms.
Graph transformations by analyzing shifts and stretches.

Formulas

Arc Length: $s = r\theta$
Exponential Growth: $P(t) = P_0(1+r)^t$
Logarithmic Equation: $\log_b(x) = y \iff b^y = x$
Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$
Trigonometric Identity: $\sin^2(\theta) + \cos^2(\theta) = 1$

Key Terms

Exponential Function: A function of the form $f(x) = a^x$, where $a$ is a positive constant not equal to 1.
Logarithm: The exponent to which a base must be raised to produce a given number.
Trigonometric Function: Functions that relate an angle of a right-angled triangle to the ratios of its side lengths (e.g., sine, cosine, tangent).
Arc Length: The distance along the curved line making up an arc.
Function Transformation: Changes applied to a parent function, such as shifts, stretches, and reflections.

Real World Examples

Population growth of a city.: Modeled using exponential functions.
Radioactive decay.: Calculated using exponential decay formulas.
Compound interest in savings accounts.: Demonstrates exponential growth.
Tracking identity theft reports.: Involves analyzing data trends over time.
Bacterial culture growth.: Often follows exponential patterns.

Timeline

Year 0: Initial population or value is established.
Year 1: Population or value grows according to the model.
Year 5: Value after 5 years is calculated using the growth formula.
Present: Data collected on reports, populations, or measurements.
Future: Projections made based on established growth or decay rates.

People

Ted: Owner of a comic book collection experiencing value increase.
Scientists: Study bacterial cultures and their growth rates.
Economists: Analyze population growth and economic trends.

Quiz

Which function best fits the scatter plot?: f(x) = 2^x
Solve for x: $\sqrt{x+5} = 19$: x = 196
Evaluate log base 0.25.: 1/32