WER - Working with Exponential Relationships (Overview)
Working with Exponential Relationships
Introduction
Now that you understand what an exponential function is, we are going to work on analyzing exponential models. 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
- How do I build an exponential function that models a relationship between two quantities?
- How do I build new functions from existing functions?
- How do I use exponential models and functions to interpret and analyze real world situations?
- Why are geometric sequences functions?
- How can we use real-world situations to construct and compare exponential models and solve problems?
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.
Explicit Expression - A formula that allows direct computation of any term for a sequence a 1 , a 2 , a 3 , . . . , a n , . . . .
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.
Recursive Formula - A formula that requires the computation of all previous terms to find the value of an .
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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