/*! 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 23 Graph the function and its refle... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 23

Graph the function and its reflection about the x-axis on the same axes. $$f(x)=\frac{1}{2}(4)^{x}$$

Short Answer

Expert verified
Plot \( f(x) = \frac{1}{2}(4)^x \) and its reflection \( g(x) = -\frac{1}{2}(4)^x \) on the same graph.

Step by step solution

01

Understand the Function

The given function is \( f(x) = \frac{1}{2}(4)^x \). This is an exponential function where the base is 4 and the multiplier is \( \frac{1}{2} \). As \( x \) increases, \( f(x) \) grows exponentially.
02

Calculate Key Points for the Function

Choose key values for \( x \) like -1, 0, 1, and 2 to find corresponding \( f(x) \) values. Compute:- \( f(-1) = \frac{1}{2} (4)^{-1} = \frac{1}{8} \)- \( f(0) = \frac{1}{2} (4)^{0} = \frac{1}{2} \)- \( f(1) = \frac{1}{2} (4)^{1} = 2 \)- \( f(2) = \frac{1}{2} (4)^{2} = 8 \)
03

Graph the Function

Plot the points \((-1, \frac{1}{8}), (0, \frac{1}{2}), (1, 2), (2, 8)\) on a coordinate plane. Connect these points smoothly because the function is continuous, forming an upward-growing curve.
04

Reflect the Function About the X-Axis

To find the reflection of \( f(x) \) over the x-axis, take \( -f(x) \). The reflected function is \( g(x) = -\frac{1}{2}(4)^x \). Calculate the reflection points using key \( x \) values:- \( g(-1) = -\frac{1}{8} \)- \( g(0) = -\frac{1}{2} \)- \( g(1) = -2 \)- \( g(2) = -8 \)
05

Graph the Reflected Function

Plot the points corresponding to \( g(x) \) as \((-1, -\frac{1}{8}), (0, -\frac{1}{2}), (1, -2), (2, -8)\). Connect these points smoothly, creating a downward-growing curve. These points are a mirror image of their counterparts in \( f(x) \) along 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.

Exponential Functions
Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They appear in the general form of \( f(x) = a \, b^x \) where \( a \) is a constant multiplier, and \( b \) is the base of the exponential. In our example, \( f(x) = \frac{1}{2} (4)^x \), the base is \( 4 \), meaning that for every one-unit increase in \( x \), the function's value quadruples.
These functions grow or decay exponentially. If the base \( b \) is greater than 1, the function grows rapidly as \( x \) increases. If \( b \) is between 0 and 1, it decays. Here, the base is 4, so \( f(x) \) increases sharply.
  • Exponential growth implies a rapid increase in values as \( x \) increases.
  • A constant multiplier, like \( \frac{1}{2} \) here, scales the function vertically.
  • Understanding how exponential functions work is essential for graphing them accurately.
Graphing Functions
Graphing is the process of plotting a function's values on a coordinate plane. For exponential functions like \( f(x) = \frac{1}{2} (4)^x \), we plot several key points to understand the function's behavior. We start by selecting strategic \( x \) values, calculate corresponding \( f(x) \) values, and plot these coordinates.
The points calculated in our solution, \((-1, \frac{1}{8}), (0, \frac{1}{2}), (1, 2), (2, 8)\), provide a rough outline of the curve. Connecting these points smoothly helps visualize how the curve behaves between plotted points:
  • It's crucial to choose both positive and negative \( x \) values for a complete view.
  • Plotting helps identify the exponential curve's direction and steepness.
  • A continuous curve suggests there's more than just these plotted points.
Coordinate Plane
The coordinate plane is a two-dimensional space formed by two perpendicular number lines, the x-axis, and y-axis. We use it to represent and visualize mathematical functions graphically. Every point on this plane is defined by an \( (x, y) \) pair, making it a powerful tool for interpreting equations and their reflections.
Understanding the axes is crucial. The x-axis is horizontal, while the y-axis is vertical. When plotting functions like \( f(x) \) and its reflection, each point is placed according to its \( x \) and \( f(x) \) values. This clear visual representation makes complex algebraic functions more comprehensible:
  • Positive x-values are to the right, negative ones to the left of the y-axis.
  • Positive y-values are above, while negative ones are below the x-axis.
  • The origin, \( (0,0) \), is where these axes intersect.
Function Transformation
Function transformation involves changing a graph's position or shape without altering its essential properties. In our exercise, we see a simple transformation: reflection.
Reflecting a function like \( f(x) = \frac{1}{2}(4)^x \) over the x-axis amounts to multiplying its output by -1. This transformation yields \( g(x) = -\frac{1}{2}(4)^x \). The reflection changes the upward curve of \( f(x) \) to a downward one:
  • The reflection makes the function curve a mirror image along the x-axis.
  • This concept can apply to both function reflections and rotations.
  • Understanding transformations is essential for manipulating and interpreting different function types.

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

Use this scenario: A biologist recorded a count of 360 bacteria present in a culture after 5 minutes and 1,000 bacteria present after 20 minutes. Rounding to six significant digits, write an exponential equation representing this situation. To the nearest minute, how long did it take the population to double?

Round all answers to the nearest thousandth. Recall the formula for calculating the magnitude of an earthquake, \(M=\frac{2}{3} \log \left(\frac{S}{S_{0}}\right)\) . One earthquake has magnitude 3.9 on the MMS scale. If a second earthquake has 750 times as much energy as the first, find the magnitude of the second quake. Round to the nearest hundredth.

For the following exercises, sketch the graph of the indicated function. $$g(x)=\log (6-3 x)+1$$

For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places. $$ \log _{\frac{1}{2}}(4.7) $$

Refer to Table. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 7.5 & 6 & 5.2 & 4.3 & 3.9 & 3.4 & 3.1 & 2.9 \\ \hline \end{array} $$ Use the LOGarithm option of the REGression feature to fi d a logarithmic function of the form \(y=a+b \ln (x)\) that best fits the data in the table.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

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.