Properties of Exponents and Graphing Exponential Functions

Introduction

You've learned about two types of functions: linear & quadratic; and now it is time for one more! Exponential Functions model a different type of growth, one that grows by a rate, rather than a constant value like a linear function. We are going to start by learning about properties of exponents and how to graph exponential functions. You will learn to analyze exponential functions so that later on you can compare them to the other types of functions you've learned about!

Essential Questions

What are the properties of exponents and how do I use them to simplify expressions?
What is an exponential function? How do I graph an exponential function?
What are the effects on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative)?
How do I find the value of k given the graphs?
What are the key features of an exponential function, and how do I identify them?
Why is the concept of a function important, and how do I use function notation to show a variety of situations modeled by functions?
What is the average rate of change of an exponential function?

Key Terms

The following key terms will help you understand the content in this module.

Algebra - The branch of mathematics that deals with relationships between numbers, utilizing letters and other symbols to represent specific sets of numbers, or to describe a pattern of relationships between numbers.

Average Rate of Change - The change in the value of a quantity by the elapsed time. For a function, this is the change in the y-value divided by the change in the x-value for two distinct points on the graph.

Coefficient - A number multiplied by a variable in an algebraic expression.

Continuous - Describes a connected set of numbers, such as an interval.

Discrete - A set with elements that are disconnected.

Domain - The set of x-coordinates of the set of points on a graph. The set of x-coordinates of a given set of ordered pairs. The value that is the input in a function or relation.

End Behaviors - The appearance of a graph as it is followed farther and farther in either direction.

Equation - A number sentence that contains an equals symbol.

Exponential Function - A nonlinear function in which the independent value is an exponent in the function, as in y = ab ^x.

Expression - Any mathematical calculation or formula combining numbers and/or variables using sums, differences, products, quotients including fractions, exponents, roots, logarithms, functions, or other mathematical operations.

Horizontal Translation - A shift in which a plane figure moves horizontally.

Interval Notation - A notation representing an interval as a pair of numbers. The numbers are the endpoints of the interval. Parentheses and/or brackets are used to show whether the endpoints are excluded or included.

Irrational Number - A number whose decimal form is nonterminating and nonrepeating. Irrational numbers cannot be written in the form a/b, where a and b are integers (b cannot be zero). So all numbers that are not rational are irrational.

Natural Numbers - The set of numbers 1, 2, 3, 4, ... Also called counting numbers.

Ordered Pair - A pair of numbers, (x, y), that indicate the position of a point on a Cartesian plane.

Range -The set of y-coordinates of the set of points on a graph. The set of y-coordinates of a given set of ordered pairs. The set of all possible outputs of a function or relation.

Rational Number - A number that can be written as a/b where a and b are integers, but b is not equal to 0.

Real Numbers - All the rational and irrational numbers that is, all of the numbers that can be expressed as decimals.

Reflection - A transformation that "flips" a figure over a mirror or reflection line.

Variable - A letter or symbol used to represent a number.

Whole Numbers - The set of numbers 0, 1, 2, 3, 4,....

x-intercept - The point where a line meets or crosses the x-axis.

y-intercept - The point where a line meets or crosses the y-axis.

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