Problem 33 - College Algebra

Question

In Exercises \(25-34,\) 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 utility to confirm your hand-drawn graphs. $$ g(x)-2 \cdot 2^{x} $$

Step-by-Step Solution

Answer

The graph of the function \(g(x) = 2 \cdot 2^{x}\) is a vertical stretch by a factor of 2 of the graph of the function \(f(x) = 2^{x}\). It has the same asymptote at \(y = 0\) or the x-axis and has a domain of (-∞, ∞) and a range of (0, ∞). Confirming with a graphing utility will ensure the accuracy of the hand-drawn graph.

Step 1: Graphing the base function

Start by graphing the function \(f(x) = 2^{x}\). Note the characteristics of this graph: it crosses the y-axis at (0,1), it's increasing, and it asymptotes to the x-axis.

Step 2: Transforming the base function

Apply a vertical stretch by a factor of 2 to graph the function \(g(x) = 2 \cdot 2^{x}\). This means every y-coordinate of the function \(f(x) = 2^{x}\) will be multiplied by 2.

Step 3: Equations of the asymptotes

Identify the asymptote of the graph of the function \(g(x) = 2 \cdot 2^{x}\). For exponential functions of this type, the asymptote is the line \(y = 0\). This can be seen as the x-axis.

Step 4: Domain and range of the function

Identify the domain and range of the function \(g(x) = 2 \cdot 2^{x}\). The domain of this function is the set of all real numbers, so it can be written as (-∞, ∞). The range of the function is \(y > 0\), hence, the range can be written as (0, ∞). This is because exponential functions with a positive base always have a range of \(y > 0\).

Step 5: Confirmation with a graphing utility

Confirm the hand-drawn graph with a graphing utility. Ensure that the graph has the correct shape, and that the asymptote is properly located at \(y = 0\).

Key Concepts

Transformations of Exponential Functions

When you work with exponential functions like the base graph of \(f(x) = 2^x\), applying transformations changes the graph's shape and position. To graph \(g(x) = 2 \cdot 2^x\), we start by stretching the base function vertically. Imagine pulling the graph upwards so that each point on the base graph moves twice as far from the x-axis. This doesn't affect the x-values, so the function still crosses the y-axis at the same point (0,1) but the y-values are all doubled. Transformations like these include shifts left or right, up or down, and reflections across the axes, allowing for a wide variety of shapes from a single base function.

For instance, adding a number inside the base function's exponent, like \(2^{x+c}\), shifts the graph horizontally, while adding a number outside the function, like \(2^x + c\), shifts it vertically. Multiplying the function by a negative value can flip it over an axis. Understanding these transformations is crucial to graphing exponential functions accurately.

Asymptotes of Exponential Functions

An asymptote is a line that a graph approaches but never actually touches or crosses. For exponential functions like \(f(x) = 2^x\), the asymptote is typically on one of the axes. With \(g(x) = 2 \cdot 2^x\), the asymptote remains the horizontal line \(y = 0\), also known as the x-axis. These functions get closer and closer to the asymptote as x moves toward negative infinity but never cross it.

No matter how you transform the exponential function, if it's in the form of \(a \cdot b^x\) (where 'a' and 'b' are constants and 'b' is greater than 1), the asymptote remains unchanged. It's key to mark the asymptote when graphing because it defines the boundary beyond which the graph doesn’t extend.

Domain and Range of Exponential Functions

The domain of an exponential function is the set of all possible inputs, or x-values, for the function. For functions like \(g(x) = 2 \cdot 2^x\), the domain is all real numbers, expressed as \((-\infty, \infty)\). It means you can plug in any number for 'x', and the function will provide an output.

The range, however, is the set of all possible outputs or y-values. Because exponential functions grow without bound, they never produce negative values when the base is greater than 1. The range of \(g(x) = 2 \cdot 2^x\) is all positive numbers, written as \((0, \infty)\). It's important to identify the domain and range correctly to understand the behavior of the function fully.

Graphing Utility Confirmation

While hand-drawing graphs gives great insight into the function's behavior, using a graphing utility serves as confirmation. A graphing calculator or software can quickly plot \(g(x) = 2 \cdot 2^x\) and verify the transformations you've applied. It’s an excellent way to check that your asymptote at \(y = 0\) and the stretch factor of 2 are correctly depicted in your sketch.

When you compare the utility's graph to your own, make sure the x-intercepts, y-values, and general shape are in agreement. This step helps catch any errors and solidifies your understanding of how exponential functions behave graphically. For educators, confirming graphs with a utility can also be a tool to demonstrate accuracy and reliability in graphing to students.