Problem 32 - Algebra and Trigonometry

Question

begin by graphing \(f(x)=2^{x} .\) Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes Use the graphs to determine each function's domain and range. If applicable, use a graphing urility to confirm your hand-drawn graphs. $$ g(x)=2^{-x} $$

Step-by-Step Solution

Answer

The function \(g(x) = 2^{-x}\) is the reflection of the function \(f(x) = 2^{x}\) over the y-axis and has the same domain(all real numbers) and range (\(y > 0\)). The horizontal asymptote for both functions is the line at y = 0. The hand-drawn graphs and the graphs plotted using a graphing utility should look the same.

Step 1: Graph Initial Function

Start by graphing \(f(x) = 2^x\). The graph is an increasing exponential curve, passing through the point (0,1), and the line y = 0 is the horizontal asymptote.

Step 2: Transformation

Next transform the graph to get the function \(g(x) = 2^{-x}\), which is obtained from \(f(x)\) by reflecting the function over the y-axis. It is decreasing, passes through the point (0,1), and the line y = 0 is still the horizontal asymptote.

Step 3: Domain and Range

The domain of an exponential function like \(f(x)\) or \(g(x)\) is the set of all real numbers. The range of \(f(x) = 2^x\) is \(y > 0\), and the range of our transformed \(g(x) = 2^{-x}\) is also \(y > 0\). For both functions, the horizontal asymptote is the line y = 0.

Step 4: Verify with Graphing Tool

Use a graphing utility, such as Desmos or a graphing calculator, to confirm the graphs. The graphs should match with the hand-drawn graphs. Make sure your final graph reflects the properties and transformations noted earlier in steps 1-3.

Key Concepts

Graph Transformations

Graph transformations are techniques used to modify the basic graph of a function. In the case of exponential functions, transformations can include shifts, reflections, stretches, and compressions. For the function \(f(x) = 2^x\), the transformation to obtain \(g(x) = 2^{-x}\) involves reflecting the graph across the y-axis. This reflection flips the exponential growth into a decay, making the graph of \(g(x)\) decreasing. This demonstrates how reflections can change the direction of an exponential graph while maintaining its shape and asymptote.

Domain and Range

Understanding the domain and range of functions is crucial in graph analysis. The domain of an exponential function like \(f(x) = 2^x\) or \(g(x) = 2^{-x}\) includes all real numbers. This is because exponential functions are defined for any real number input without restriction.
The range of \(f(x) = 2^x\), such as \(g(x) = 2^{-x}\), is \(y > 0\). Exponential functions never take zero or negative values due to their definition. It's important to grasp these concepts to accurately represent and understand the graphical behavior of exponential functions.

Asymptotes

Asymptotes are lines that a graph gets infinitely close to but never touches. For both \(f(x) = 2^x\) and \(g(x) = 2^{-x}\), the horizontal asymptote is the line \(y = 0\). This means as \(x\) approaches negative or positive infinity, the graph approaches the line \(y = 0\) but never reaches it. Recognizing asymptotes helps in sketching graphs more accurately, as it indicates the behavior of the function at the extremes.

Graphing Utilities

Graphing utilities are valuable tools for confirming the behavior of functions. They can help visualize transformations and verify properties like domain, range, and asymptotes. Tools like Desmos or graphing calculators can plot \(f(x) = 2^x\) and \(g(x) = 2^{-x}\) accurately, showing the reflections and how they affect the graph. Using these tools aids in understanding and verifies the accuracy of hand-drawn graphs, ensuring they reflect all changes and properties found in the analysis.