Problem 50 - 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. $$g(x)=e^{x+1}+2$$

Step-by-Step Solution

Answer

The graph of the function \(g(x)=e^{x+1}+2\) appears to increase from left to right with the horizontal asymptote at \(y=2\).

Step 1: Construct a Table of Values

Use a graphing utility and pick values for \(x\) and compute the corresponding values for \(g(x)\). For example, -2, -1, 0, 1 and 2 could be chosen. These values will give a general shape of the function.

Step 2: Sketch the Graph

Plot the points from the table of values on a graph and sketch a curve through these points. It should appear to be increasing from left to right.

Step 3: Identify Asymptotes

Observe the behavior of the curve at extremes. If the curve appears to get closer to a line but never cross it as \(x\) approaches positive or negative infinity, that line is an asymptote. In this case, \(y=2\) is the horizontal asymptote because as \(x\) heads towards negative infinity, the y-values head towards 2.

Key Concepts

Understanding Exponential Functions

An exponential function is a type of mathematical function that involves a constant base raised to a variable exponent. In our exercise, the function given is \(g(x) = e^{x+1} + 2\). This means that the variable \(x\) is in the exponent, and the base is \(e\), which is a mathematical constant approximately equal to 2.718. Exponential functions like this one are widely used in various fields due to their growth properties that can model populations, radioactive decay, and financial calculations.
One key element of exponential functions is their general behavior: they tend to increase or decrease rapidly, depending on the function's base and the sign of the exponent. In the given function, the term \(e^{x+1}\) represents the exponential portion, and adding 2 shifts the entire graph vertically upward by 2 units. This shift is critical for determining the graph's characteristics, including the asymptote.

Exploring Asymptotes

Asymptotes are lines that a graph approaches but never actually touches. They are like invisible barriers that guide the shape of a graph. For our function \(g(x) = e^{x+1} + 2\), we find the asymptote by looking at the behavior of the graph as \(x\) becomes very large or very small.
In this particular case, the horizontal asymptote is at \(y=2\). Here's how you can identify this:

• As \(x\) moves toward negative infinity, the term \(e^{x+1}\) (which is always positive) approaches zero, and thus \(g(x)\) approaches 2.
• Therefore, the function never actually reaches \(y=2\), it only gets closer and closer.

The asymptote's presence means that no matter how far our \(x\)-values go, the graph will not cross this line, providing a boundary for the behavior of \(g(x)\).

Constructing a Table of Values

To graph a function effectively, constructing a table of values is a helpful step. This begins by selecting some input values for \(x\), then calculating the corresponding \(g(x)\) values. This was simply outlined in the original solution by choosing \(x\) values such as -2, -1, 0, 1, and 2.
Here's how you create this table:

• Choose values for \(x\). It's often helpful to select both negative and positive values.
• Calculate \(g(x)\) for each \(x\) using the formula: \(g(x)=e^{x+1}+2\).
• Record these \(g(x)\) values in the table alongside the corresponding \(x\) values.

For example:

• When \(x=-2\), \(g(x)=e^{-1}+2\).
• When \(x=0\), the calculation becomes \(g(0)=e^1+2\).

This table visually provides a set of points you can plot on a graph, aiding in sketching the function accurately.

Sketching the Graph

Once the table of values is ready, the next step is sketching the graph. This involves plotting the \(x, g(x)\) pairs from your table onto a graph, which gives you a picture of how the function behaves.
Here's a simple way to approach it:

• Start by drawing a set of axes, labeling them \(x\) and \(y\).
• Plot each point from your table onto the graph.
• Observe any patterns — with exponential functions, you'll typically see a curve that increases or decreases rapidly.
• Join the plotted points smoothly with a curve, considering the asymptotic behavior as you draw. Here, the curve will rise steadily from left to right because \(e^{x+1}\) grows larger as \(x\) increases.

Remember, sketching is about approximating the shape. The goal is to visualize how the function behaves based on calculations and identified characteristics, such as the asymptote at \(y=2\).