Sketch the graphs of the given functions on the same axes. y=2^x, y=3^x, and y=4^x

Name: Problem 27, Chapter 5: Applied Calculus for the Managerial, Life, and Social Sciences
Uploaded: 2021-05-13T07:57:52-08:00
Duration: 3 min 13 s
Channel: Priyanka Sadarangani
Description: Sketch the graphs of the given functions on the same axes. y=2^x, y=3^x, and y=4^x

step 1 Okay, in this question three functions are given that is y equals to 2 raise to the power x Okay

Sketch the graphs of the given functions on the same axes....

Key Concepts

Exponential Functions

Exponential functions are mathematical functions in which a constant base is raised to a variable exponent. They are characterized by their rapid increase or decrease, depending on whether the base is greater than 1 (growth) or between 0 and 1 (decay). Their graphs are smooth curves that are always positive, have a horizontal asymptote (commonly the x-axis), and pass through the point (0,1), reflecting the property that any nonzero number raised to the power of zero equals one.

Growth Rates

The growth rate of an exponential function is determined by its base; a larger base results in more rapid growth. This means that among several exponential functions with bases greater than 1, the one with the highest base will increase faster as the input variable increases. Understanding this concept is crucial for comparing how different exponential functions behave relative to each other across the domain.

Graphing Functions on the Same Axes

When graphing multiple functions on the same axis, it is important to observe their relative positions and behavior. Plotting exponential functions together allows for visual comparison of their growth rates and intercepts, showcasing characteristics such as the common point through which all exponential functions pass (0,1) and how their curves diverge as the input increases. This skill is essential in understanding function transformation and comparing different mathematical models.

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