Exponential Function

An exponential function is a mathematical function of the form f(x) = a * b^x, where a and b are constants and b is a positive real number not equal to 1.

Key Concepts:

Base (b): The base of an exponential function is the constant raised to the power of x.
Exponent (x): The exponent represents the power to which the base is raised.
Growth and Decay: Exponential functions can model both growth (when b > 1) and decay (when 0 < b < 1).

Graphing Exponential Functions:

To graph an exponential function, you can plot points or use transformations of the basic exponential function f(x) = b^x. When b > 1, the graph increases from left to right. When 0 < b < 1, the graph decreases from left to right.

Properties of Exponential Functions:

Domain: The domain of an exponential function is all real numbers.
Range: The range of an exponential function depends on the value of b. When b > 1, the range is all positive real numbers. When 0 < b < 1, the range is all positive real numbers less than 1.
Asymptote: The x-axis (y = 0) is a horizontal asymptote for the graph of an exponential function.

Common Exponential Functions:

Some common exponential functions include:

Exponential growth: f(x) = a * b^x, where b > 1
Exponential decay: f(x) = a * b^x, where 0 < b < 1

Real-World Applications:

Exponential functions can be used to model population growth, radioactive decay, compound interest, and more.

Study Guide:

Understand the basic form of an exponential function: f(x) = a * b^x
Identify the base and exponent in an exponential function
Graph exponential functions and understand their behavior based on the value of the base
Explore real-world applications of exponential functions
Practice solving problems involving exponential growth and decay

Exponential functions are an important topic in mathematics and have wide-ranging applications in various fields. Understanding the properties and behaviors of exponential functions can help in analyzing and solving real-world problems.

◂Math Worksheets and Study Guides Sixth Grade. Proportions/Equivalent Fractions

Study Guide

Proportions/Equivalent Fractions

Worksheet/Answer key

Proportions/Equivalent Fractions

Worksheet/Answer key

Proportions/Equivalent Fractions

Worksheet/Answer key

Proportions/Equivalent Fractions

Create And Print more Fractions worksheets with Comparing Fractions, Equivalent Fractions, and Simplifying Fractions

The resources above cover the following skills:

Number and Operations (NCTM)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Understand and use ratios and proportions to represent quantitative relationships.

Compute fluently and make reasonable estimates.

Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

Grade 6 Curriculum Focal Points (NCTM)

Number and Operations: Connecting ratio and rate to multiplication and division

Students use simple reasoning about multiplication and division to solve ratio and rate problems (e.g., 'If 5 items cost $3.75 and all items are the same price, then I can find the cost of 12 items by first dividing $3.75 by 5 to find out how much one item costs and then multiplying the cost of a single item by 12'). By viewing equivalent ratios and rates as deriving from, and extending, pairs of rows (or columns) in the multiplication table, and by analyzing simple drawings that indicate the relative sizes of quantities, students extend whole number multiplication and division to ratios and rates. Thus, they expand the repertoire of problems that they can solve by using multiplication and division, and they build on their understanding of fractions to understand ratios. Students solve a wide variety of problems involving ratios and rates.