/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 Sketch a graph of the given func... [FREE SOLUTION] | 91影视

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Chapter 0: Problem 21

Sketch a graph of the given function. $$f(x)=e^{2 x}$$

Short Answer

Expert verified
The graph of \(f(x)=e^{2x}\) starts at the point (0,1) and ascends sharply as \(x\) increases. As \(x\) decreases, the graph approaches the x-axis but never touches it. The graph lies entirely above the x-axis.

Step by step solution

01

Understand the basic shape of the function

The function represents an exponential function, whose base form is \(f(x)=e^x\). The basic shape of an exponential function is ascending to the right if the exponent x is positive and descending to the right if x is negative. The graph is always above the x-axis, but never touches or crosses it.
02

Effect of the parameter

The parameter '2' in \(f(x)=e^{2x}\) modifies the rate at which the function grows. Because '2' is positive, the function will grow twice as fast. As x increases, \(f(x)\) will increase more sharply than the base function \(f(x)=e^x\). As x decreases, \(f(x)\) will decrease more sharply towards 0.
03

Sketch the graph

Starting from the y-axis where \(x=0\), denote the point (0,1) since \(e^{2*0}=1\). As x increases, the function \(f(x)=e^{2x}\) will ascend more sharply than the normal exponential function. As \(x\) decreases, the curve approaches the x-axis but never crosses or touches it. The graph is above the x-axis for all \(x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Growth
Exponential growth occurs when a quantity increases at a rate proportional to its current value. In the function \( f(x) = e^{2x} \), we see this behavior through the rapid increase of \( f(x) \) as \( x \) becomes larger. With exponential growth:
  • The rate of growth is constantly increasing.
  • The initial value grows larger at an accelerating pace.
  • This type of function models many natural phenomena, such as population growth or compounding interest.
In \( f(x) = e^{2x} \), the '2' in the exponent multiplies the growth rate compared to the base function \( f(x) = e^x \), causing the growth to happen at an even faster rate.
Remember, as \( x \) increases, \( f(x) \) rises exponentially above the x-axis, which is a hallmark feature of exponential growth. Understanding this concept will help you analyze patterns and predict behavior in similar exponential functions.
Graphing Techniques
Creating a precise graph of an exponential function involves understanding its unique characteristics. For \( f(x) = e^{2x} \):
  • Start by identifying the y-intercept. For any \( f(x) = e^{kx} \), where \( k \) is a constant, the y-intercept is at point (0, 1) since \( e^{0} = 1 \).
  • Observe the curvature. Exponential graphs curve upwards, since they grow faster as \( x \) increases.
  • Plot a few key points to guide the sketch. For example, plug \( x = 1 \) into \( f(x) = e^{2x} \) to find another point: \( e^2 \).
These points and the overall shape of the curve will guide your graphing efforts. A common challenge is ensuring the curve does not touch or cross the x-axis, as the function will never equal zero. Practice plotting such exponential functions to build confidence in your graphing skills.
Transformations of Functions
Transformations are modifications to the basic form of a function that change its shape or position. For exponential functions like \( f(x) = e^{2x} \), transformations occur when variables and constants alter the growth rate or direction:
  • Vertical Stretch/Shrink: The coefficient of \( x \) in the exponent determines the rate of growth. In \( f(x) = e^{2x} \), the "2" causes a vertical stretch, making the function grow more rapidly.
  • Horizontal Shifting: Adding or subtracting inside the exponent, such as \( f(x) = e^{2(x - 1)} \), shifts the graph horizontally.
  • Vertical Shifting: Adding or subtracting a constant outside the exponential term can move the graph up or down.
Understanding these transformations allows you to anticipate how changes in the function's formula will alter its graph. This knowledge lets you quickly grasp how modified functions will appear without needing to plot every point.

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