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Chapter 4: Problem 25

Use a graphing utility to graph the function. \(y=-2^{x}\)

Short Answer

Expert verified
The graph of the function \(y=-2^{x}\) is a line decreasing from left to right, approaching the x-axis but never touching it. Due to the negative sign, the function is reflected across the x-axis compared to the base function \(2^{x}\).

Step by step solution

01

Use a Graphing Utility

To graph the function \(y=-2^{x}\), start by choosing a graphing utility. Many free options are available online, for example Desmos, GraphFree, or GeoGebra. These allow for an easy input of your function and quickly display the graph.
02

Input the Function

Enter the function \(y=-2^{x}\) into the graphing utility. Make sure to put parentheses around the exponent to ensure that the utility understands the power applies to the full number 2, and not just the sign of the number.
03

Analyze the Graph

After inputting the function, the graphing utility will display a graph. The function \(y=-2^{x}\) is a decreasing function because of the negative sign, so you will observe a line decreasing from left to right, getting closer to the x-axis but never touching it. This is because the value of the function approaches zero as x increases, but will never reach zero due to the nature of exponential functions.

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Key Concepts

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

Exponential Functions
Exponential functions play a significant role in mathematics, especially when modeling growth and decay processes. In general, an exponential function is expressed in the form \(y = a \cdot b^{x}\), where:
  • \(a\) is a constant, representing the initial value or the y-intercept.
  • \(b\) is the base of the exponential, which determines the growth or decay factor.
  • \(x\) is the variable or exponent.
In our function, \(y=-2^{x}\), \(a\) is implicitly -1, and \(b\) is 2. Importantly, the negative before the base results in a reflection over the x-axis, making the function decreasing rather than growing. Consequently, as \(x\) increases, the output y-value approaches zero, but never quite reaches it.
Graphing Functions
Graphing functions is an essential skill in understanding their behavior and characteristics visually. To graph a function like \(y = -2^{x}\), a graphing utility is incredibly helpful. Popular tools include Desmos, GeoGebra, and GraphFree, which allow users to input equations and view graphs instantly.When graphing, make sure to:
  • Input the function carefully, especially paying attention to exponents and signs.
  • Use parentheses to ensure operations are computed correctly. For example, inputing \(-2^{(x)}\) vs \(-2^{x}\).
  • Adjust viewing windows to see important features of the graph, such as intercepts or asymptotic behaviors.
These tools provide interactive visualizations, enabling you to manipulate values and observe changes dynamically.
Analyzing Graphs
Analyzing a graph involves understanding its shape, behavior, and key features. In the case of \(y = -2^{x}\), several critical observations can be made:
  • The graph is a reflection of a typical exponential growth function across the x-axis due to the negative sign.
  • As \(x\) increases, the y-values become increasingly negative, indicating a downward trend.
  • The function approaches the x-axis as an asymptote, meaning it gets infinitely close but never touches.
  • On a graph, it will appear to come down from the left, rapidly descend, and linearly approach zero, never intercepting the x-axis.
Analyzing graphs helps provide insights into functions鈥� trends and behaviors, such as identifying intercepts, asymptotes, and changes in direction, which are integral for deeper understanding.

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Most popular questions from this chapter

graph and analyze the function. Include any relative extrema and points of inflection in your analysis. Use a graphing utility to verify your results. $$ y=x \ln x $$

Use a graphing utility to verify that the functions are equivalent for \(x>0 .\) $$ \begin{array}{l}{f(x)=\ln \sqrt{x\left(x^{2}+1\right)}} \\\ {g(x)=\frac{1}{2}\left[\ln x+\ln \left(x^{2}+1\right)\right]}\end{array} $$

Compound Interest Use a spreadsheet to complete the table, which shows the time \(t\) necessary for \(P\) dollars to triple if the interest is compounded continuously at the rate of \(r .\) $$ \begin{array}{|c|c|c|c|c|c|c|c|}\hline r & {2 \% |} & {4 \%} & {6 \%} & {8 \%} & {10 \%} & {12 \%} & {14 \%} \\ \hline t & {} & {} & {} & {} & {} \\\ \hline\end{array} $$

Solve for \(x\) or \(t\) $$ 400(1.06)^{t}=1300 $$

Use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. $$ \ln x y z $$
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