Problem 48 - Intermediate Algebra

Question

For Problems \(35-52\), graph each exponential function. $$ f(x)=2^{2 x} $$

Step-by-Step Solution

Answer

Graph the curve of \(f(x) = 2^{2x}\) using plotted points and note the exponential growth.

Step 1: Understand the Exponential Function

The function given is \( f(x) = 2^{2x} \), which is an exponential function where the base is 2 and the exponent is \(2x\). This means that for every increase in \(x\), the value of \( f(x) \) doubles raised to the power of \(2x\).

Step 2: Create a Table of Values

Select a few values for \(x\) to find corresponding \(f(x)\) values. For example:- If \(x = -1\), \( f(x) = 2^{2(-1)} = 2^{-2} = \frac{1}{4}\).- If \(x = 0\), \( f(x) = 2^{2(0)} = 2^{0} = 1\).- If \(x = 1\), \( f(x) = 2^{2(1)} = 2^{2} = 4\).- If \(x = 2\), \( f(x) = 2^{2(2)} = 2^{4} = 16\).

Step 3: Plot the Points

Using the table of values from Step 2, plot the points \((-1, \frac{1}{4})\), \((0, 1)\), \((1, 4)\), and \((2, 16)\) on a graph.

Step 4: Draw the Graph

Draw a smooth curve through all the plotted points. The curve should approach 0 as \(x\) becomes very negative and rise steeply as \(x\) becomes positive, reflecting the exponential growth of the function.

Step 5: Analyze the Graph

The graph of \(f(x) = 2^{2x}\) is an exponential growth curve that starts near zero and increases rapidly as \(x\) increases. The asymptote is the x-axis (\(y = 0\)).

Key Concepts

Graphing Exponential Functions

Graphing exponential functions like \( f(x) = 2^{2x} \) offers a visual representation of how these functions behave. To start graphing, you need to make a table of values, choosing specific \( x \) values and calculating the corresponding \( f(x) \) values. This allows you to plot points on a graph.
By doing so, you illustrate how the function changes across different \( x \) values. For instance:

• At \( x = -1 \), \( f(x) = \frac{1}{4} \)
• At \( x = 0 \), \( f(x) = 1 \)
• At \( x = 1 \), \( f(x) = 4 \)
• At \( x = 2 \), \( f(x) = 16 \)

Connecting these points with a smooth curve reveals the exponential nature of the function, particularly how it rises swiftly as \( x \) increases. It's crucial to observe the direction and steepness, which signify exponential growth.

Understanding Asymptotes

An important feature of exponential graphs is the asymptote. In \( f(x) = 2^{2x} \), the asymptote is the x-axis or \( y = 0 \). This means the function gets closer and closer to the x-axis as \( x \) moves towards negative infinity although it never actually touches it.
Asymptotes represent a boundary which the curve approaches but does not cross. They give insight into the behavior of the function at the extreme parts of its domain. In exponential functions, the asymptote often indicates the lower bound of the function’s value. Therefore, knowing where the asymptote lies helps in drawing accurate graphs and understanding behavior at boundary conditions.

Exponential Growth Explained

Exponential growth is a key characteristic of exponential functions. In \( f(x) = 2^{2x} \), as \( x \) increases, the value of the function grows rapidly. This is due to the constant base (in this case 2) being raised to increasingly larger powers.
This growth pattern is not linear. Instead, it accelerates as \( x \) gets larger. What makes exponential growth compelling is how quickly functions can increase. Understanding these dynamics can help you predict how the function behaves over larger ranges of \( x \). This is typical in various real-world applications such as populations growing exponentially or compound interest in finance.

Creating a Table of Values

A table of values is a crucial step in graphing any function, including exponential ones like \( f(x) = 2^{2x} \). By selecting a series of \( x \) values and calculating their corresponding \( f(x) \) values, you lay the groundwork for plotting these points on a graph.
For example, by computing:

• \( f(-1) = \frac{1}{4} \)
• \( f(0) = 1 \)
• \( f(1) = 4 \)
• \( f(2) = 16 \)

you can accurately depict the function's behavior. Each computed coordinate provides a specific point on the graph, making the transformation from abstract equation to visual more comprehensible. This methodical approach ensures precision and clarity when sketching the graph of the function.