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

Name: Problem 29, Chapter 5: Applied Calculus for the Managerial, Life, and Social Sciences
Uploaded: 2021-05-13T08:03:14-08:00
Duration: 3 min 11 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

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.

Transcript

00:01 Okay, in this question three functions are given that is y equals to 2 raise to the power negative x y raised to 3 raise to the power negative x and y equals to 4 raise to the power negative x and now we have to sketch the graph of this given function.

00:15 So first of all we can say 2 raise to the power x it will be 1 by 2 raise to the power x okay it will be 1 by 3 raised to the power x and it will be 1 by 4 raised to the power x and looking at these functions we can say okay i'm writing down here y equals to the power x.

00:30 Y equals to and y equals to so looking at these functions we can say these are defined for all real numbers okay and there will be a y intercept at x equals to zero y will be one for all these three functions okay and there will be no x intercept and now we will take some values of x and these function y so that we can make a sketch graph of these functions okay so first of all we will take okay just take it we will take some x equals to negative 3 negative 2 negative 1 0 1 2 and 3 so i'm writing down these points here and here it will be 1 by 2 raised to the power x okay 1 by 3 to the power x and 1 by 4 raise to the power x so we will put the values and 2 raise to the power minus x it will be 1 by 2 raise to the power x no problem and now it will be 8 okay and x equals to negative 3 1 by 2 raise to the power x the power negative 3 or we can say 2 raise to the power positive 3 that will be 8 and now 2 raise to the power positive 2 it will be 4 okay now it will be 2 and x equals to 0 it will be 1 1 and it will be 1 okay and now x equals to 1 1 by 2 raise to the power 1 it will be 1 by 2 it will be 1 by 4 and 1 by 8 and now we will do same for 1 by 3 raise to the power x so it will be start from 27 then 9 then 3 it is 1 already and now 1 by 3 1 by 9 and 1 by 20 and same for 1x4 raised to the power x 64 it will be 16 and 4...

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