Exponential Functions and Sequences
Lessons in this chapter
About Exponential Functions and Sequences
Exponential Functions and Sequences reviews exponent and radical properties, then introduces an entirely new function family that grows (or shrinks) by repeated multiplication rather than repeated addition.
Properties of Exponents and Radicals and Rational Exponents review and extend the exponent rules from 8th grade, connecting radical notation (like a square root) to fractional exponents.
Exponential Functions introduces functions of the form y = a·bˣ, where the output is repeatedly multiplied by a constant factor b, and Exponential Growth and Decay applies this to real situations like population growth or radioactive decay, depending on whether b is greater than or less than 1.
Solving Exponential Equations has kids solve for an unknown exponent, while Geometric Sequences and Recursively Defined Sequences close the chapter by treating a sequence with a constant ratio between terms as a discrete version of an exponential function.
The seven lessons in this chapter each open as an Algebra 1 worksheet in the page, including Properties of Exponents, Exponential Growth and Decay and Recursively Defined Sequences. Questions are scored as you answer them, which helps on the growth and decay set, where the whole answer turns on whether the base is above or below 1. All seven are free to try, with no login needed for your first 3.
Frequently asked questions
Properties of Exponents and Radicals and Rational Exponents come first, extending the exponent rules to negative and fractional powers. Exponential Functions, Exponential Growth and Decay and Solving Exponential Equations then cover graphs that multiply by a fixed factor each step, the situations they model, and how to solve for an unknown exponent. Geometric Sequences and Recursively Defined Sequences close the chapter by writing the same repeated multiplication as a sequence, either in terms of its position or in terms of the term before it. The chapter draws on the high school Number and Quantity (N), Algebra (A) and Functions (F) conceptual categories.
An exponential function has the form y = a·bˣ, where the output is repeatedly multiplied by a constant factor b for every increase of 1 in the input: unlike a linear function, which repeatedly ADDS a constant.
Growth happens when the base b is greater than 1 (the output keeps increasing), and decay happens when b is between 0 and 1 (the output keeps decreasing toward zero): both use the exact same y = a·bˣ form.
A rational exponent like x^(1/2) means the same thing as a radical, √x: the denominator of the exponent tells you which root to take, and the numerator tells you what power to raise the result to.
If both sides can be written with the same base, set the exponents equal to each other and solve; otherwise, techniques introduced later (like logarithms) become necessary for a full solution.
A geometric sequence has a constant RATIO between consecutive terms (each term is the previous term times the same number), the discrete version of an exponential function the same way an arithmetic sequence is the discrete version of a linear one.