Problem 43 - 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. $$y=3^{x-2}+1$$

Step-by-Step Solution

Answer

The graph of the function \(y=3^{x-2}+1\) rises sharply, and skews to the right. It has a horizontal asymptote at y = 1.

Step 1: Create a Table of Values

Choose a set of x-values and evaluate the function \(y=3^{x-2}+1\) at each. For instance, you can use x-values like 0, 1, 2, 3, 4 and calculate the corresponding y-values.

Step 2: Plotting the Function

Next, graph the function using the table of values. Each (x, y) pair represents a point in the graph. Plot each point and fill the spaces between the points to give an accurate representation of the graph of the function.

Step 3: Identify The Asymptotes

An asymptote is a line that the graph approaches but never touches. By observing the graph, we can see that there is a horizontal asymptote at y = 1. This is because as x approaches negative infinity, the \(3^{x-2}\) term will approach 0 and the function will be closer to y = 1.

Key Concepts

Asymptotes in Graphs

Understanding the presence of asymptotes is critical when graphing exponential functions. An asymptote is a line that a graph approaches indefinitely but never actually reaches, no matter how far the graph is extended in either direction.

For the exponential function given by the equation \( y = 3^{x-2} + 1 \), you can identify the horizontal asymptote through observation and analysis. Since an exponential function's rate of growth or decay is proportional to its current value, the function \( y = 3^{x-2} \) will grow significantly as \( x \) increases and will approach zero as \( x \) decreases. But because of the '+1' at the end of the function, the graph's y-value will asymptotically approach 1, never going below it. This means that the line \( y = 1 \) is a horizontal asymptote.

Constructing Tables of Values

Creating a table of values is an initial step that can help visualize an exponential function before plotting it. To construct a table, select a range of x-values, preferably including both positive and negative numbers to capture the behavior of the function across different intervals. For each x-value, calculate the corresponding y-value using the function \( y = 3^{x-2} + 1 \).

This process involves performing calculations for each chosen x-value and captures snapshots of what the graph looks like at those specific points. By carefully choosing points around the horizontal asymptote, you can gain a better understanding of how the graph behaves near this boundary. Remember, including values that show the approach to the asymptote on both sides enhances the accuracy of the graph sketch.

Plotting Functions

With the values from your table, you can now begin plotting the function. On a coordinate plane, mark each (x, y) pair from your table as a point. Connect these points smoothly; remember that with exponential functions, the graph should continuously curve away from the horizontal asymptote.

As you plot the points based on the table of values, you'll notice the rapid increase or decrease between the y-values. The characteristic 'J' shape (or a flipped 'J' in the case of a negative exponent) of an exponential function graph is distinct and should become apparent. Ensuring the curve never crosses the horizontal asymptote at \( y = 1 \) is key to an accurate representation of the function.