Sketch the graphs of the given functions on the same axes. y=4^0.5 x, y=4^x, and y=4^2 x

Name: Problem 31, Chapter 5: Applied Calculus for the Managerial, Life, and Social Sciences
Uploaded: 2021-05-13T08:23:05-08:00
Duration: 3 min 20 s
Channel: Priyanka Sadarangani
Description: Sketch the graphs of the given functions on the same axes. y=4^0.5 x, y=4^x, and y=4^2 x

step 1 Okay, in this question three functions are given, that is, I'm writing down here, Y equals to 4

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

Key Concepts

Comparing Exponential Growth Rates

Comparing the growth rates of exponential functions is crucial when analyzing multiple functions on the same graph. Functions with larger coefficients in the exponent will grow much faster than those with smaller coefficients. This comparison allows for an analysis of how subtle changes in the exponent's multiplier can significantly alter the function's behavior over the same domain.

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. They are characterized by rapid growth or decay and always pass through the point (0,1) when the function is in the form y = a^(bx) with a > 0 and a ? 1. This concept is central to understanding how changes in the exponent affect the overall function behavior.

Graphing Exponential Functions

Graphing exponential functions involves identifying key features such as the y-intercept, the asymptote (usually the x-axis), and the rate of increase or decrease. By plotting these features, one can visualize how the function behaves over its domain, which is essential for comparing multiple exponential curves on the same set of axes.

Transformation of Functions

Transformations involve modifying the basic exponential function through horizontal or vertical shifts, stretches, or compressions. Changing the coefficient in the exponent, as seen in variations of y = a^(bx), results in horizontal scaling. This directly affects the steepness or rate of growth/decay of the graph, a fundamental concept in understanding how graphs are altered by different parameter values.

Transcript

Please give Ace some feedback