This document provides an overview of exponential functions including:
- Definitions of exponential functions for positive integer, rational, and irrational exponents. Conventions are established to define exponents for all real number bases and exponents.
- Properties of exponential functions including that they are continuous functions with domain of all real numbers and range from 0 to infinity. Properties involving addition, subtraction, multiplication, and division of exponents are proven.
- Examples are provided to demonstrate simplifying expressions using properties of exponents, including fractional exponents.
The document discusses various transformations that can be performed on graphs of functions, including vertical translations, stretches, and compressions. Vertical translations move the entire graph up or down by adding or subtracting a constant to the function. Stretches elongate or compress the graph vertically by multiplying the function by a constant greater than or less than 1, respectively. These transformations can be represented by modifying the original function in a way that corresponds to the geometric transformation of its graph.
The document discusses solving systems of 3 linear equations with 3 unknowns. It provides examples of using the elimination method, which involves rewriting the system as two smaller systems, eliminating the same variable from each, solving the resulting system of 2 equations for the remaining 2 variables, then substituting back into one of the original equations to find the third variable. The solution is written as an ordered triple (x, y, z). It demonstrates this process on examples and encourages practicing this method.
This document provides an overview of exponential functions including:
- Definitions of exponential functions for positive integer, rational, and irrational exponents. Conventions are established to define exponents for all real number bases and exponents.
- Properties of exponential functions including that they are continuous functions with domain of all real numbers and range from 0 to infinity. Properties involving addition, subtraction, multiplication, and division of exponents are proven.
- Examples are provided to demonstrate simplifying expressions using properties of exponents, including fractional exponents.
The document discusses various transformations that can be performed on graphs of functions, including vertical translations, stretches, and compressions. Vertical translations move the entire graph up or down by adding or subtracting a constant to the function. Stretches elongate or compress the graph vertically by multiplying the function by a constant greater than or less than 1, respectively. These transformations can be represented by modifying the original function in a way that corresponds to the geometric transformation of its graph.
The document discusses solving systems of 3 linear equations with 3 unknowns. It provides examples of using the elimination method, which involves rewriting the system as two smaller systems, eliminating the same variable from each, solving the resulting system of 2 equations for the remaining 2 variables, then substituting back into one of the original equations to find the third variable. The solution is written as an ordered triple (x, y, z). It demonstrates this process on examples and encourages practicing this method.