Problem 34 - 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. $$ g(x)=\frac{1}{2} \cdot 2^{x} $$

Step-by-Step Solution

Answer

The graph of \(g(x)=\frac{1}{2} \cdot 2^{x}\) is similar to that of \(f(x)=2^{x}\), but vertically compressed by a factor of 1/2. The domain is all real numbers, and the range is \(y>0\).

Step 1: Graph the base function

Start by graphing the base function, \(f(x)=2^{x}\). This is an exponential function, and its graph is a upward curving line. It has a horizontal asymptote at \(y=0\).

Step 2: Apply the transformations

The given function \(g(x)=\frac{1}{2} \cdot 2^{x}\) can be obtained from the base function by a vertical shrink by a factor of 1/2. This means, each y-coordinate of the original function will be halved. Asymptotes of the function remain unchanged.

Step 3: Draw the transformed graph

Draw the graph of \(g(x)=\frac{1}{2} \cdot 2^{x}\) , which will have the same general shape as the base function \(f(x)=2^{x}\), but will be closer to the x-axis.

Step 4: Determine the domain and range

The domain of this function, similar to the base function, is all real numbers. However, the range is \(y>0\), because exponential functions never take negative values.

Step 5: Use a graphing utility to confirm

Finally, you can use a graphing utility to confirm the details of the graph drawn by hand. The graph should agree with the transformations done on the base function

Key Concepts

Exponential Function Transformations

Understanding transformations of exponential functions is crucial for visualizing their graphs correctly. Take the base function f(x) = 2^x, known for its upward curve and horizontal asymptote at y = 0. When we introduce a transformation, such as a vertical shrink, in our exercise with the function g(x) = \(\frac{1}{2} \times 2^{x}\), we are essentially scaling the y-values of the graph by a factor of \(\frac{1}{2}\).

This transformation does not affect the horizontal asymptote or the domain but alters the steepness of the graph. It's like pressing down on a spring: it still springs upward, but not as high. By understanding this concept, students can graphically represent different scenarios and equations with confidence, ensuring the transfomations reflection in their graphs.

Domain and Range of Functions

The domain and range of a function represent, respectively, the set of possible input values (x-values) and output values (y-values) for that function. For all exponential functions, including our base function f(x) = 2^x and its transformed counterpart g(x) = \(\frac{1}{2} \times 2^{x}\), the domain is always all real numbers because there is no x-value for which the function is undefined.

However, the range is cautiously tiptoed; it is always y > 0, as exponential functions never reach zero, let alone go negative. Remembering this will prevent students from assigning impossible values to these ever-growing (or shrinking) functions.

Asymptotes of Functions

An asymptote is like a boundary that the graph of a function approaches but never truly touches or crosses. In the realm of exponential functions, we often deal with horizontal asymptotes. For the base function f(x) = 2^x, this asymptote is found at y = 0. No matter how far the graph stretches along the x-axis, it maintains a respectful distance from the y-axis, as if drawn by an invisible magnetic force.

When transformations occur, like with g(x) = \(\frac{1}{2} \times 2^{x}\), the location of the horizontal asymptote doesn’t waver—it remains steadfast at y = 0. This unwavering line serves as a valuable navigational tool for students to predict how the graph behaves towards infinity, always ensuring they're on the right path.