Problem 129 - 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. $$\begin{array}{l}y_{1}=3.25 \\\y_{2}=\frac{1}{2} \ln (x+2)\end{array}$$

Step-by-Step Solution

Answer

The intersection of the two functions is approximately at the point which can be determined from the graphing utility, rounded to three decimal places.

Step 1: Graph the First Function

Plot the graph for the function \(y_{1}=3.25\). This line is a horizontal one crossing the y-axis at 3.25.

Step 2: Graph the Second Function

Next, plot the graph for the function \(y_{2}=\frac{1}{2} \ln (x+2)\). This function represents a logarithmic curve starting from \(x=-2\).

Step 3: Identify the intersection

The point of intersection is the point which lies on both graphs. By analyzing from the graphing utility, an approximation of the intersection can be made to the third decimal place.

Step 4: Determine the x-coordinate of the intersection point

To find the x-coordinate, see where the two graphs intersect on the x-axis.

Step 5: Determine the y-coordinate of the intersection point

The y-coordinate of the intersection point is obtained by substituting the value of x from the previous step into either of the given equations.

Key Concepts

Intersection of Graphs

When we talk about the intersection of graphs, we refer to the point where two different graphs meet or cross each other. It's like finding that special point on a map where two roads cross.
Understanding this concept is important because the intersection can tell us a lot, like common solutions or equilibrium points in various practical problems.
Here's a simple way to find the intersection:

• Graph each function separately. In our exercise, these are the horizontal line of the first function and the curved line of the logarithmic function.
• Observe where these lines meet on the graph. This meeting point is the intersection.

Ideally, using a graphing utility, you can zoom in and get a closer approximation of the intersection point's coordinates. Make sure to jot these down! This is often done with mathematical software which makes it easy to see exactly where these graphs cross each other on a screen. In our case, the graphing utility will help us approximate this to three decimal places.

Logarithmic Function

Logarithmic functions might seem complex, but they are very powerful tools in mathematics. Here, the function we are examining, \(y = \frac{1}{2} \ln(x+2)\), is a type of logarithmic function.
Let's break it down:

• The component \(\ln(x+2)\) represents the natural logarithm of \(x+2\). The natural logarithm deals with the logarithmic base \(e\), where \(e\) is a special number, approximately 2.718.
• The \(\frac{1}{2}\) outside the logarithm is a multiplier that affects the steepness of the curve, essentially scaling the function by 0.5.

Logarithmic functions like this one usually start from the x-value that sets the argument of the log to zero. Hence, this graph starts from \(x = -2\). As \(x\) increases, the function's value increases slowly forming a curve. It's essential to understand the behavior of this curve when trying to find intersections.

Approximation

Approximation is a super useful concept in mathematics, especially when exact numbers are either impossible to find or not necessary. In this exercise, we're asked to find an approximate intersection point of the two graphs.
But how do we approximate? The answer lies in our graphing utility:

• Once both functions are plotted, visually analyze where they meet.
• Use the tool to zoom in and read off the point. You might not get an exact number but a very close one.
• Round this number to three decimal places as instructed.

Approximations help us deal with complex problems by finding practical solutions quickly. They're especially handy in real-world applications where exact numbers might not be available or needed. Always remember, the key is being as close as possible to the real value.