Problem 126 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate the point of intersection of the graphs. Round your result to three decimal places. $$\begin{array}{l}y_{1}=4 \\\y_{2}=3^{x+1}-2\end{array}$$

Step-by-Step Solution

Answer

The intersection point of the graphs of the two functions \(y_1 = 4\) and \(y_2 = 3^{x+1} - 2\) is approximately at \((x, y) = (0.287, 4)\), where the x-coordinate has been rounded to three decimal places.

Step 1: Plot Both Functions

Use a graphing utility to plot both the functions. The first function \(y_1 = 4\) is a horizontal line at y=4. The second function \(y_2 = 3^{x+1} - 2\) is an exponential function shifted down by 2 units and to the left by 1 unit. The vertical asymptote is at x = -1.

Step 2: Find the Point of Intersection

The point of intersection is the point on the graph where the two functions meet. In this case, the intersection represents the solution to the equation \(y_1 = y_2\), or \(4 = 3^{x+1} - 2\). By visually inspecting the graphs or using the graphing utility's intersection feature, you can approximate this point.

Step 3: Round to Three Decimal Places

Once the intersection point has been found, round the x-coordinate to three decimal places as per the instructions in the exercise.

Key Concepts

Intersection of Graphs

When you study graphs of functions, one important concept is finding where two graphs intersect. An intersection is a point on the graph where two different functions meet. In mathematical terms, it’s the specific set of coordinates \((x, y)\) that satisfies both equations simultaneously.

In this exercise, you're dealing with two functions: a horizontal line \(y_1 = 4\) and an exponential curve \(y_2 = 3^{x+1} - 2\). The point where these two graphs intersect corresponds to the solution of the equation \(4 = 3^{x+1} - 2\). To find this intersection point, you can set each equation equal to each other and solve for \(x\).

Graphing this relationship helps to provide a visual perspective, allowing you to easily see where the graphs meet. Remember, this point of intersection can be found using various methods, such as analytical solving or employing a graphing utility for precision.

Graphing Utility

A graphing utility is an incredibly useful tool for visualizing mathematical functions and their interactions. It can be a calculator, computer software, or an app that plots functions quickly and accurately. In this exercise, using a graphing utility simplifies the process of finding intersections between graphs.

Here is a quick guide to using a graphing utility for this exercise:

• Input the functions to be plotted. For example, enter \(y_1 = 4\) and \(y_2 = 3^{x+1} - 2\).
• Observe the graph output, noting where the curves meet.
• Use the intersection or analysis feature to find the exact intersection point.

By entering the expressions directly into the utility, you can easily visualize the problem and trust the calculated point of intersection. Many graphing tools even allow you to zoom in to get a more precise view, helping you to refine your results down to the needed decimal places.

Exponential Function

Exponential functions, like \(y_2 = 3^{x+1} - 2\), play a crucial role in mathematics, modeling growth and decay scenarios. They have a general format of \(a^{(x + c)} + d\), where \(a\) is a positive constant, \(c\) shifts the graph horizontally, and \(d\) shifts it vertically.

For the given function, \(3^{x+1} - 2\), the base is \(3\), indicating the exponential nature. The graph has been shifted left by 1 unit (due to \(+1\)) and down by 2 units (due to \(-2\)). Additionally, such functions exhibit a vertical asymptote that influences their appearance. For this example, the vertical asymptote occurs when \(x o -\infty\), though it's not defined explicitly in the exercise.

Understanding these transformations is key to predicting the behavior of exponential functions and locating their intersections with other graphs. Always remember, the steeper the curve, the more rapidly it rises or falls as you move along the x-axis.