Problem 60 - Algebra and Trigonometry

Question

The Height of the Graph of an Exponential Function Your mathematics instructor asks you to sketch a graph of the exponential function $$ f(x)=2^{x} $$ for \(x\) between 0 and \(40,\) using a scale of 10 units to one inch. What are the dimensions of the sheet of paper you will need to sketch this graph?

Step-by-Step Solution

Answer

X-axis: 4 inches; Y-axis: impractical with linear scale due to height being \(2^{40}\).

Step 1: Understanding the Function

The function given is an exponential function, specifically, it's \( f(x) = 2^x \). Exponential functions grow very quickly, and it’s important to consider this when selecting the size of paper needed to plot the graph.

Step 2: Determine the Range on the X-axis

You've been asked to graph the function for \( x \) between 0 and 40. With a scale of 10 units to one inch, this means the x-axis will span 40 units. Hence, you will need \( 40/10 = 4 \) inches for the x-axis.

Step 3: Calculate the Extent on the Y-axis

To determine the paper height needed, evaluate the function at x = 40: \( f(40) = 2^{40} = 1,099,511,627,776 \). This is a very large y-value. You’ll need enough space to represent this value on the y-axis.

Step 4: Convert the Y-axis into Inches

With a y-value as large as \( 2^{40} \), representing this would be impractical on paper if using a linear scale of 10 units per inch. For practical plotting, either a smaller scale per inch or a logarithmic scale would generally be used. Given the question's constraint, the full paper height would require looking at custom scaling adjustments or higher mathematical tools than linear inches.

Step 5: Conclusion and Paper Dimensions

While the x-axis requires 4 inches, the y-axis with current constraints severely restricts practical plotting. Typically, an infinite height is required as \( y = 2^{40} \). For a realistic approach or dimensions, logarithms or limited sections should be used, but those aren't specified here directly.

Key Concepts

Graphing

Graphing exponential functions like \( f(x) = 2^x \) can seem daunting at first. However, by plotting even a few points, you can start to see the rapid rise that characterizes these functions. Start by calculating several key points, such as \((0, 1)\), \((1, 2)\), \((2, 4)\), and so on.
Connect these points with a smooth curve to illustrate the exponential growth. Remember, an exponential graph continuously increases and never dips below the x-axis.
When preparing your graph, pay attention to the enormous increase that happens even between \( x = 10 \) and \( x = 40 \), where values really skyrocket.

Function Properties

Understanding the properties of exponential functions like \( f(x) = 2^x \) helps in grasping how to graph them effectively. Exponential functions are defined as functions where a constant base is raised to a variable exponent. In this case, the base is 2.

• Exponential growth means that the function will increase rapidly as x increases.
• The output (y-value) is never negative.
• The domain of the function is all real numbers. However, practical graphing often focuses on a specific range of x-values, like 0 to 40 in this example.
• The range is only positive numbers, as a result of the base being a positive number.

Properties like these are crucial for evaluating growth and understanding how the function will behave on your graph.

Scale Adjustment

With exponential functions, scale adjustment becomes a critical consideration. In the given task, using a linear scale of 10 units per inch initially suggests using a 4-inch paper width for x-values from 0 to 40.
However, the main challenge arises in accommodating the y-values.

• Exponential growth quickly moves to very large numbers, making regular scaling impractical.
• To combat this, consider alternative methods, such as decreasing the unit per inch on the y-axis or even using a logarithmic scale.
• Creative adjustments help represent this function on a tangible scale, as \( 2^{40} \) is absolutely massive.

Approaches like these are key to effectively visualizing and printing exponential graphs on paper.

Exponential Growth

Exponential growth refers to increasing growth rates that occur in exponential functions like \( f(x) = 2^x \). It's a fantastically fast rate of increase, which becomes evident even with slight changes in x.
Here's why exponential growth is significant:

• It starts slowly, as seen with initial values like \( f(0) = 1 \) and \( f(1) = 2 \).
• It then shoots up drastically, with \( f(10) = 1024 \), showing how each small increase in x multiplies the value in a massive way.
• This rapid increase continues, making high x-values challenging to graph traditionally within reasonable space.

Understanding exponential growth is crucial for visual representation and dealing with real-world phenomena that follow similar patterns, such as population growth or viral spread.