/*! 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 18 Sketch the graph of the function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 18

Sketch the graph of the function.\(y=e^{-x / 2}\)

Short Answer

Expert verified
The graph of the function \(y=e^{-x / 2}\) will be a downward-sloping curve above the x-axis. It starts at the point (0, 1) and gradually approaches the x-axis, but never crosses it, as x tends towards positive infinity. As x tends towards negative infinity, the function increases towards positive infinity.

Step by step solution

01

Understand the behavior of the function

For the function given as \(y=e^{-x / 2}\), the base is \(e\), and it is raised to the power \(-x / 2\). Since the exponent is a negative value divided by 2, the graph will decay as x increases from left to right. Furthermore, because \(e^{-x / 2}\) is always greater than 0, the x-axis is a horizontal asymptote.
02

Identify points on the graph

You can plot some key points to help with sketching the graph by substituting values for \(x\) into \(y=e^{-x / 2}\). When \(x=0\), \(y=e^{0}=1\). As \(x\) becomes larger, the function \(e^{-x / 2}\) approaches zero. Also, as \(x\) approaches negative infinity, \(y\) approaches positive infinity.
03

Sketch the graph

Now, draw a smooth curve that passes through the points you found in the previous step. The graph will approach but never cross the x-axis (the horizontal asymptote). The function starts at a high point on the y-axis and decays as x increases, which forms a downward-sloping curve above the x-axis.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Graphing Exponential Functions
Exponential functions are a fundamental concept in mathematics, particularly when dealing with growth and decay. In our specific function, we have the equation \(y=e^{-x / 2}\).
This is an exponential function where the base is Euler's number \(e\), known for its unique mathematical properties. To graph an exponential function efficiently, it's crucial to understand how the value of the exponent affects the graph shape.
  • **Base and Exponent:** The base \(e\) is raised to the power of \(-x / 2\). As \(x\) values increase or decrease, the changes in the exponent directly influence the value of \(y\).
  • **Plotting Points:** A practical way to start the graph is by calculating specific points using different \(x\) values. For instance, substituting \(x = 0\) gives \(y = 1\).
  • **Overall Shape:** Once multiple points are plotted, they provide a guide for drawing the curve. Remember, for exponential decay, the curve will slope downwards if the exponent is negative, like in our case.
This visually represents the behavior of the function as it moves along the x-axis, allowing us to better interpret the underlying mathematical relationships.
Asymptotic Behavior
Asymptotic behavior in exponential functions is essential to understand, as it dictates how the function behaves as it approaches certain limits. The asymptote is a line that the graph approaches but never touches.
  • **Horizontal Asymptote:** In our function \(y=e^{-x / 2}\), the horizontal asymptote is the line \(y = 0\) (the x-axis). Since an exponential function like this never reaches zero, the curve will keep approaching the x-axis but won鈥檛 cross it.
  • **Long-Term Behavior:** As \(x\) increases towards positive infinity, \(y\) gets closer to zero. Conversely, as \(x\) approaches negative infinity, \(y\) grows unbounded in the positive direction.
  • **Graph Interpretation:** Understanding the horizontal asymptote helps us predict the behavior of the graph in the longer term, ensuring a more accurate sketching and interpretation of exponential decay processes.
The asymptotic properties provide insight into the function's ultimate destiny, shaping how we graph and understand exponential equations.
Decay of Exponential Functions
Exponential decay is a key concept when dealing with functions where the rate of change decreases over time. Our function \(y = e^{-x / 2}\) is an example of exponential decay due to its negative exponent.
  • **Decay Characteristics:** When dealing with decay, you'll observe that as \(x\) increases, the value of \(y\) decreases. This is because the exponent is negative, causing the function to reduce in value.
  • **Visual Representation:** Graphically, this decay forms a downward slope that progressively flattens as it approaches the x-axis. The curve starts high on the y-axis and falls as \(x\) grows.
  • **Practical Relevance:** Exponential decay models are frequently used in real-world applications, such as physics for radioactive decay or finance for modeling decreasing investments.
Understanding the decay aspect of exponential functions is vital as it equips you with tools to model and analyze diminishing processes over time.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Solve the differential equation. $$ \frac{d y}{d x}=\frac{1-2 x}{4 x-x^{2}} $$

Find the integral. $$ \int \frac{2}{x \sqrt{1+4 x^{2}}} d x $$

Find \(\left(f^{-1}\right)^{\prime}(a)\) for the function \(f\) and the given real number \(a\). \(f(x)=\sin x, \quad-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}, \quad a=\frac{1}{2}\)

Find the limit. $$ \lim _{x \rightarrow 0^{-}} \operatorname{coth} x $$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Show that \(f(x)=\int_{2}^{x} \sqrt{1+t^{2}} d t\) is one-to-one and find \(\left(f^{-1}\right)^{\prime}(0)\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.