EEE - Working with Exponential Relationships (Overview)
Modeling and Analyzing Exponential Expressions & Equations
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.
Many professions use exponential growth: statisticians, doctors, bankers, and city planners. Imagine your grandfather left you an heirloom guitar, and you've researched to find that it grows in value by 8% each year. If you understand exponential functions, you'll be able to write a model to describe the value of your guitar at any time!
Essential Questions
- What are the properties of exponents and how do I use them to simplify expressions?
- 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?
- How do I build an exponential function that models a relationship between two quantities?
- How do I use exponential models and functions to interpret and analyze real-world situations?
- How can we use real-world situations to construct and compare exponential models and solve problems?
Key Terms
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.
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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