Problem 28 - 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. $$ g(x)=2^{x}+2 $$

Step-by-Step Solution

Answer

The function \(g(x) = 2^x + 2\) is the graph of \(f(x) = 2^x\) shifted upward by 2 units. This shift doesn't change the domain which is \( (-\infty, \infty) \), but the range becomes \( (2, \infty) \) due to vertical shift. The horizontal asymptote is at y = 2.

Step 1: Graph the original function

Begin by graphing the original function \(f(x) = 2^x\). This function is an exponential function, with \(2^x\) as its base. The graph of this function will be an increasing curve that crosses the y-axis at \(y = 1\) and has a horizontal asymptote at y = 0. The domain for \(f(x) = 2^x\) is \( (-\infty, \infty) \) and the range is \( (0, \infty) \).

Step 2: Understand the transformation

Given function \(g(x) = 2^x + 2\) is transformed from the original function by vertically shifting all points 2 units upwards. The new graph will still be an increasing curve, but now it crosses the y-axis at y = 3 and the horizontal asymptote is at y = 2.

Step 3: Graph the transformed function

Graph the transformed function \(g(x) = 2^x + 2\) with the earlier described transformation. Notice the shift 2 units upward compared to the original graph of \(f(x) = 2^x\).

Step 4: Determine domain and range of transformed function

The domain of the transformed function \(g(x) = 2^x + 2\) is still all real numbers, so \( (-\infty, \infty) \), as x-values were not altered. Because of the upward shift, the range now starts from 2 to infinity, thus the range is \( (2, \infty) \).

Step 5: Confirm with a graphing utility

This step involves confirming the drawn graph with the help of a graphing utility, as it will provide a more accurate version of the graph, and ensure the above steps are correct and the transformations, domain and range have been understood and applied correctly.

Key Concepts

Exponential Function Transformations

Understanding exponential function transformations is crucial when you encounter a function like \( g(x) = 2^x + 2 \). This equation represents a transformation of the base function \( f(x) = 2^x \). In this case, the entire graph of the base function shifts vertically upward by 2 units.

Imagine the graph of \( f(x) = 2^x \) as a set of points. When you add 2 to the function, you're effectively adding 2 to the \( y \)-value of each point, which raises the graph's position on the coordinate plane. This operation doesn't affect the shape of the graph; it simply moves it as a whole. These transformations can include shifts, stretches, compressions, and reflections, each altering the graph in a different manner. To graph \( g(x) = 2^x + 2 \), you would take the original exponential curve and shift every point up by two units, creating a new graph that retains the same growth rate but starts from a higher value.

Domain and Range of Functions

The domain of a function is the set of all possible input values (\( x \)-values) for which the function is defined. For exponential functions like \( f(x) = 2^x \), the domain is all real numbers (\( (-fty, fty) \)) because you can raise 2 to any power.

The range, on the other hand, is the set of possible output values (\( y \)-values). For the base function \( f(x) = 2^x \), since the exponential function never touches the \( x \)-axis (horizontal asymptote at \( y = 0 \)), the range is \( (0, fty) \). When transformed to \( g(x) = 2^x + 2 \), the entire graph shifts upward, increasing all of the \( y \)-values by 2. Thus, the range of \( g(x) \) becomes \( (2, fty) \), indicating that \( g(x) \) will never produce a value less than 2.

Asymptotes of Exponential Functions

An asymptote is a line that the graph of a function approaches but never actually touches. Exponential functions like \( f(x) = 2^x \) have horizontal asymptotes, which in this case is the \( x \)-axis, or \( y = 0 \).

The presence of an asymptote is related to the behavior of the graph at extreme values of \( x \). As \( x \) becomes very large, the function approaches infinity, and as \( x \) becomes very negative, the function's value gets closer and closer to the asymptote without ever reaching it. When the function is transformed to \( g(x) = 2^x + 2 \), the horizontal asymptote also shifts upwards by the same amount, changing from \( y = 0 \) to \( y = 2 \). This new asymptote serves as a boundary below which the graph will never go, no matter how far you extend it to the left.

Graphing Utility Use in Algebra

A graphing utility is an essential tool in visualizing algebraic functions and verifying manual calculations. When graphing functions such as exponential transformations, a graphing utility can provide a precise and clear depiction of the function's behavior.

After graphing by hand, you can use a graphing utility to confirm the shape, position, asymptotes, domain, and range of your graph. The accurate visuals from a graphing utility can help you understand subtleties in the function's behavior that might be missed during manual graphing. A common use is to check for transformation accuracy; in this case, confirming that the graph of \( g(x) = 2^x + 2 \) is indeed a vertical shift of the base graph \( f(x) = 2^x \) by 2 units upward. Another benefit is the ability to zoom in or out to observe the behavior near the asymptote or at extreme values of \( x \), making it an invaluable aid in algebraic studies.