Modeling Radical, Exponential and Logarithmic Functions Module Overview

This module consists of two parts, each focusing on different functions. Part one covers radical functions, while part two explores exponential and logarithmic functions. Each part will be assessed separately to evaluate your understanding and progress.

Part 1: Radical Functions Overview

Radical functions are functions that involve radicals, which can be written as rational exponents.

Radical Expressions are expressions involving radicals, and when written with radicals this is known as radical form. Radical expressions can also be written with rational exponents (known as exponential form). These radicals contain an index (sometimes also referred to as a "root"). Roots and powers are opposite operations. These similarities are also in the operations of addition and subtraction and the operations of multiplication and division.

For an integer, n, greater than 1, if $b^n=a$ , then b is an nth root of a. An nth root of a, written as $LaTeX: \sqrt[n]{a}$ $\sqrt[n]{a}$ , where n is the index of the radical. If n is odd, then a has one real nth root. If n is even, then a has two real nth roots a > 0, one real nth root if a = 0, and no real nth roots if a < 0. When we are graphing radical functions, we will specifically explore the square roots ( $LaTeX: y=\sqrt[]{x}$ $y=\sqrt[]{x}$ ) and cube roots ( $LaTeX: y=\sqrt[3]{x}$ $y=\sqrt[3]{x}$ ). In this unit, we'll explore some more specifics of radical functions, operations on radical functions (written in radical form and in rational exponent form), solving radical functions and inequalities, and then graphing radical functions and inequalities.

Essential Questions

What are various properties of nth roots?
What are various properties of rational exponents?
How is a radical (rational exponent) equation solved?
Why are all solutions not necessarily the solution to an equation?
How can extraneous solutions be identified?
How are square root and cube root functions graphed?

Radical Functions 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.

Coefficient - A number multiplied by a variable.

Equation - A number sentence that contains an equality symbol.

Expression - A mathematical phrase involving at least one variable and sometimes numbers and operation symbols.

Extraneous Solution(s) - A solution of the simplified form of the equation that does not satisfy the original equation.

Index of a Radical - The " n " in the nth root of a, written as equation image indicator $LaTeX: \sqrt[n]{a}$ $\sqrt[n]{a}$ .

Inequality - Any mathematical sentence that contains the symbols > (greater than), < (less than), ≤ (less than or equal to), or ≥ (greater than or equal to).

Polynomial - A mathematical expression involving the sum of terms made up of variables to nonnegative integer powers and real-valued coefficients.

Radical Function - A function containing a root. The most common radical functions are the square root and cube root functions,

equation image indicator $LaTeX: f\left(x\right)=a\sqrt[]{x-h}+k\:and\:f\left(x\right)=a\sqrt[3]{x-h}+k$ $f\left(x\right)=a\sqrt[]{x-h}+k\:and\:f\left(x\right)=a\sqrt[3]{x-h}+k$

Reciprocal - Two numbers whose product is one. For example, equation image indicator $LaTeX: \frac{m}{n}\cdot\frac{n}{m}=1$ $\frac{m}{n}\cdot\frac{n}{m}=1$ .

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

Part 2: Exponential and Logarithmic Functions Overview

The applications of exponential and logarithmic functions are vast and varied. The following are just some of the applications and jobs of exponential and logarithmic functions: savings and investment accounts, loans for cars, loans for homes, nuclear power and radioactivity, forensics, sound technician, studying earthquakes, statistician working with population growth, SAT scores, etc.

Exponents (also known powers) may be written as the number of times a base is used as a factor. They may be written as a logarithm when you need to solve for the power. The graphs of exponential and logarithmic functions show the domain, the intercepts, and the end behavior of the function. Exponential and logarithmic functions are inverse functions of each other, and the x-values and the y-values may be exchanged for each other. In modeling real-world situations, we often need to solve these functions. The steps are similar to solving other algebra equations, but usually require changing from one form to the other. They are often used to calculate things like the time to double, triple (an investment or a population), or halve (radioactive half-life) a quantity.

Essential Questions

How are exponential growth functions graphed?
How are exponential growth functions applied?
How are exponential decay functions graphed?
How are exponential decay functions applied?
How is Euler's number "e" used in exponential and logarithmic functions?
How are logarithmic functions graphed and applied?
How are logarithmic properties applied?
How are exponential and logarithmic equations solved?

Exponential and Logarithm Key Terms

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

Exponential Function - A function of the form equation image indicator $LaTeX: A=a\cdot b^{x-h}+k,\:where\:a,\:h,\:and\:k$ $A=a\cdot b^{x-h}+k,\:where\:a,\:h,\:and\:k$ are real numbers, b > 0, and a and b are ≠ 1.

Exponential Growth Function - A function of the form equation image indicator $LaTeX: A=a\cdot b^{x-h}+k$ $A=a\cdot b^{x-h}+k$ where b > 1.

Growth Factor - The base number "b" with a value b > 1 in a function of the form equation image indicator $LaTeX: A=a\cdot b^{x-h}+k$ $A=a\cdot b^{x-h}+k$ where b > 1.

Asymptote - An asymptote is a line or curve that approaches a given curve arbitrarily closely. A graph never crosses a vertical asymptote, but it may cross a horizontal or oblique asymptote.

Exponential Decay Function - A function of the form equation image indicator $LaTeX: A=a\cdot b^{x-h}+k$ $A=a\cdot b^{x-h}+k$ where 0 < b < 1.

Decay Factor - The base number "b" with a value 0 < b < 1 in a function of the form equation image indicator $LaTeX: A=a\cdot b^{x-h}+k$ $A=a\cdot b^{x-h}+k$ where 0 < b < 1.

Natural Base e - Euler's number e with the approximation of 2.718…

Common Logarithm - A logarithm with a base of 10. A common logarithm is the exponent, a, such that equation image indicator LaTeX: 10^a=b $10^a=b$ . The common logarithm of x is written log x. For example, log 100 = 2 because LaTeX: 10^2=100 $10^2=100$ .

Natural Logarithm - A logarithm with a base of e. lnb is the exponent, a, such that equation image indicator LaTeX: e^a=b $e^a=b$ . The natural logarithm of x is written lnx and represents $LaTeX: \log_ex$ $\log_ex$ . For example, ln 8 = 2.0794415… because $LaTeX: e^{2.0794415}=8$ $e^{2.0794415}=8$ .

Compound Interest Formula - A method of computing the interest, after a specified time, and adding the interest to the balance of the account. Interest can be computed as little as once a year to as many times as one would like. The formula is equation image indicator $LaTeX: A=P\left(1+\frac{r}{n}\right)^{nt}$ $A=P\left(1+\frac{r}{n}\right)^{nt}$ where A is the ending amount, P is the principal or initial amount, r is the annual interest rate, n is the number of times compounded per year, and t is the number of years.

Continuous Compound Interest Formula - Interest that is, theoretically, computed and added to the balance of an account each instant. The formula is equation image indicator $LaTeX: A=Pe^{rt}$ $A=Pe^{rt}$ , where A is the ending amount, P is the principal or initial amount, r is the annual interest rate, and t is the time in years.

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