Problem 49 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$f(x)=2+e^{x-5}$$

Step-by-Step Solution

Answer

The graph of the function \(f(x) = 2 + e^{x-5}\) is an upward-shifted exponential curve with a horizontal asymptote at \(y = 2\).

Step 1: Understand the Function

Firstly, it's essential to recognize that the function \(f(x) = 2 + e^{x-5}\) is an exponential function that has been shifted up by 2 units and to the right by 5 units.

Step 2: Construct a Table of Values

Choose some x-values to evaluate the function. Make sure to include \(x = 5\), which is where the shift has occurred. For each x-value, substitute it into the function to find the corresponding y-value.

Step 3: Sketch the Graph

Using the table values, plot the points on the coordinate grid. Draw a smooth curve through the plotted points, extending the graph on both ends according to the behavior of an exponential function.

Step 4: Identify Asymptotes

As the x-values increase without bound, the function will go to positive infinity due to the exponential part. As the x-values decrease without bound, the function will go to 2. Therefore, there is a horizontal asymptote at \(y = 2\).

Key Concepts

Exponential Function Graph

Understanding the graph of an exponential function like f(x) = 2 + e^{x-5} is crucial in learning its characteristics and behavior. Exponential functions have distinct shapes that can be recognized easily. They generally rise or fall at increasingly rapid rates and don't have any x-intercepts. However, with the addition or subtraction of terms, these functions can be shifted up, down, left, or right from their 'normal' position.

When graphing the function f(x) = 2 + e^{x-5}, we're dealing with a 'shifted' exponential function - it has been moved up by 2 units due to the '+2' and translated to the right by 5 units because of the exponent (x-5). This rightward shift means the part of the graph that typically rises steeply will now do so to the right of where it would for the function e^x.

The graph will still maintain the characteristic of increasing rapidly as x increases, but the entire graph will be situated above the horizontal line y = 2. This information gives us a head start before even plotting points: our graph will never cross or touch the line y = 2, and it will pass close to the y-axis but at a point higher than 2.

Constructing a Table of Values

Starting with a table of values is a systematic approach to graphing functions, especially if you are not using graphing technology. To construct a table of values for our function, f(x) = 2 + e^{x-5}, you'll need to choose various x-values and then compute the corresponding y-values. This process outlines the shape of the graph piece by piece.

To make a comprehensive table, include x-values less than, equal to, and greater than the shift point x = 5. This gives you a view of the graph in multiple regions. For instance:

• At x = 5, the function simplifies to f(5) = 2 + e^0 = 3.
• Choosing values of x greater than 5 will show the rapid increase of the function's value due to the exponential term e^{x-5}.
• For x-values less than 5, the power on e becomes negative, causing the values to approach the horizontal asymptote from above.

Plot these points on a coordinate grid and connect them with a smooth, continuous curve that reflects the exponential nature of the function.

Identifying Asymptotes

An asymptote is a line that the graph of a function approaches but never actually reaches. For the function f(x) = 2 + e^{x-5}, this behavior can be seen as x decreases without limit. In this case, the function approaches the line y = 2. This line, y = 2, is known as a horizontal asymptote and it signifies the value that the function nears as x heads towards negative infinity.

To properly identify asymptotes, it's important to recognize the behavior of the function on both ends of the x-axis. As x becomes very large, the term e^{x-5} dominates, causing the function's value to shoot up towards positive infinity, and thus we don't have an asymptote in that direction.

However, as x becomes very negative, the exponential term gets closer and closer to zero (since e raised to a negative power gives a very small positive number), and the function's value gets closer and closer to 2. This flattening-out effect to the horizontal line y = 2 is what defines our horizontal asymptote, guiding us on the expected behavior of the graph as we move towards extreme values of x.