/*! 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 491 Construct the graph of \(\mathrm... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 20: Problem 491

Construct the graph of \(\mathrm{y}=3^{\mathrm{x}}\).

Short Answer

Expert verified
The given function \(y = 3^{x}\) is an exponential function with a base of 3. The key properties of exponential functions indicate that the x-axis is a horizontal asymptote, and the function increases for x > 0 and decreases for x < 0. Selecting a few x-values such as -2, -1, 0, 1, and 2, we calculate their corresponding y-values as 1/9, 1/3, 1, 3, and 9, respectively. After plotting these points on the coordinate plane, connect them smoothly to form the graph. The graph should be getting closer to the x-axis as x decreases and gradually become steeper as x increases. Finally, label the axes, plotted points, and the horizontal asymptote for the complete graph.

Step by step solution

01

Identify the base of the exponential function

The given function is of the form 'y = a^x', where 'a' is the base and 'x' is the exponent. In this case, the base 'a' is 3.
02

Identify important properties of exponential functions

For an exponential function, the x-axis (y=0) is a horizontal asymptote, meaning the graph approaches but never intersects the x-axis. For exponents x > 0, the exponential function increases, whereas, for x < 0, it decreases.
03

Calculate some points on the graph

We will select a few x-values and calculate their corresponding y-values using the given function, y=3^x. Note that you can calculate more points if you want a more detailed graph. \[ \begin{aligned} &\text{For } x=-2: &y &= 3^{-2} = \frac{1}{9} \\ &\text{For } x=-1: &y &= 3^{-1} = \frac{1}{3} \\ &\text{For } x=0: &y &= 3^{0} = 1 \\ &\text{For } x=1: &y &= 3^{1} = 3 \\ &\text{For } x=2: &y &= 3^{2} = 9 \end{aligned} \]
04

Plot the calculated points

Plot the points on the coordinate plane: (-2,1/9),(-1,1/3), (0,1), (1,3), and (2,9).
05

Draw the graph

Connect the plotted points smoothly, forming an increasing curve when x > 0 and a decreasing curve when x < 0. The graph should get closer to the x-axis as x approaches the negative values but never intersect it. The graph will appear steeper as x > 0 due to the rapid increase in the exponential function. For the final graph, label the axes, the calculated points, and the horizontal asymptote (x-axis). The graph of the function y=3^x is now constructed.

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.

Exponential Function Properties
Exponential functions are interesting and unique. They have several key properties that set them apart from other types of functions. Let's take a closer look at these properties:

  • The graph of an exponential function is continuous and smooth, without any breaks or sharp corners.
  • Exponential functions exhibit rapid growth or decay as the value of the exponent changes.
  • For positive base values greater than 1, the function will grow very quickly for positive exponents and will approach zero as the exponent becomes negative.
  • There is always a horizontal asymptote (a line that the graph approaches but never touches). For the function \( y = a^x \), this asymptote is the x-axis.

These properties make exponential functions ideal for modeling real-world processes like population growth, radioactive decay, and interest calculations.
Base of Exponential Functions
The base of an exponential function is a critical component because it affects how the function behaves. In our example, the base is 3, as seen in the equation \( y = 3^x \).

Here's why the base is important:

  • The base determines the rate of growth or decay of the function. A larger base will grow more quickly when \( x \) is positive, and decay faster when \( x \) is negative.
  • For a base between 0 and 1, the function will decay as \( x \) increases. For bases greater than 1, the function grows as \( x \) increases.
  • The base must always be positive, as negative or zero bases do not produce valid exponential functions.

Understanding the role of the base will help you predict and understand the shape and behavior of the graph.
Horizontal Asymptote
A horizontal asymptote is a line that the graph of the function approaches but never quite reaches or crosses. In the context of exponential functions,
the horizontal asymptote is usually the x-axis, which corresponds to \( y = 0 \).

Here's what to remember about horizontal asymptotes in exponential functions:

  • Since exponential functions diminish infinitely but never actually zero out for negative infinity, the x-axis is the horizontal asymptote for functions like \( y = a^x \).
  • The closer \( x \) gets to negative infinity, the closer the value of \( y \) gets to 0, but \( y \) will never be exactly 0.
  • This asymptotic behavior can be seen by plotting points further along the negative x-axis, confirming the function's behavior as \( x \) decreases further.

Recognizing the horizontal asymptote helps in predicting the graph's end behavior and can be useful in sketching accurate graphs quickly.
Plotting Points
Plotting points is a practical method to help visualize exponential functions, especially when you want to graph them accurately. By selecting specific x-values and calculating the corresponding y-values, you can plot these points to form the graph.

Here's how to effectively plot points for \( y = 3^x \):

  • Select a range of x-values, both negative and positive, to understand the full behavior of the function.
  • Calculate the y-values using the formula \( y = 3^x \). For instance, at \( x = -2 \), the y-value is \( \frac{1}{9} \).
  • Mark these points on the coordinate plane accurately to ensure a precise representation.

By plotting these points, like in the example with \( x = -2, -1, 0, 1, 2 \), you create a framework to visualize how the function behaves, making it easier to sketch the curves later on.
Graphing Techniques
Graphing techniques are essential to accurately represent the characteristics of exponential functions on a coordinate plane. Here's a simple guide to efficiently graph \( y = 3^x \):

  • Start by plotting your calculated points, such as \((-2, \frac{1}{9})\), \((-1, \frac{1}{3})\), and so forth.
  • Connect these points smoothly with a curve. Ensure that the curve is continuous, without sharp angles or disruptions.
  • Note the rapid increase of the graph as x becomes positive and the gradual approach towards the horizontal asymptote (the x-axis) as x becomes negative.
  • Label the graph clearly with axes, points, and the equation of the function.

Using these graphing techniques will help you create an accurate and visually clear representation of the exponential function, aiding in understanding and interpretation.

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 equation xlog x = 100x.

Write \(5^{3}=125\) in logarithmic form.

Evaluate \(\left(\log _{10} 12\right) /\left(\log _{10} 5\right)\).

Find the values of the following logarithms: a) \(\log _{10} 10\) b). \(\log _{10} 100\) c). \(\log _{10} 1\) d) \(\log _{10} 0.1\) e). \(\log _{10} 0.01\)

Express the logarithm of 7 to the base 3 in terms of common logarithms.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Geometry

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.