Problem 51 - College Algebra with Modeling and Visualization

Question

Sketch a graph of \(y=f(x)\) $$ f(x)=\left(\frac{1}{2}\right)^{x} $$

Step-by-Step Solution

Answer

Graph decreases, y-axis intercept at (0,1), approaches x-axis as x increases.

Step 1: Understanding the Function

The function given is an exponential function, defined as \(f(x) = \left(\frac{1}{2}\right)^{x}\). Exponential functions of the form \(b^x\), where \(0 < b < 1\), decrease as \(x\) increases. In this function, \(b = \frac{1}{2}\), indicating that the graph will depict a decreasing exponential behavior.

Step 2: Identifying Key Points

We need key points to sketch the graph. Start by computing values for chosen \(x\) values: - At \(x = 0\), \(f(0) = \left(\frac{1}{2}\right)^{0} = 1\).- At \(x = 1\), \(f(1) = \left(\frac{1}{2}\right)^{1} = \frac{1}{2}\).- At \(x = 2\), \(f(2) = \left(\frac{1}{2}\right)^{2} = \frac{1}{4}\).- At \(x = -1\), \(f(-1) = \left(\frac{1}{2}\right)^{-1} = 2\).These points will be used to sketch the curve.

Step 3: Graphing the Points

Plot the identified points on the graph: - Plot \((0, 1)\), the y-intercept.- Plot \((1, \frac{1}{2})\).- Plot \((2, \frac{1}{4})\).- Plot \((-1, 2)\).Ensure the graphing paper or software you're using has a clear grid to accurately position the points.

Step 4: Sketching the Graph

With the points plotted, draw a smooth curve through them. The curve should fall as \(x\) increases, approaching \(y = 0\) asymptotically, indicating that the graph never touches the x-axis but gets infinitely close to it in the positive x-direction. It rises sharply as \(x\) becomes more negative.

Key Concepts

Exponential Decay

Exponential decay refers to a process where quantities decrease at a rate proportional to their current value. In the context of our function \(f(x) = \left( \frac{1}{2} \right)^{x}\), it signifies how the function's value reduces as \(x\) increases. This reduction occurs because our base, \(\frac{1}{2}\), is less than 1.

Here's how exponential decay manifests:

• The value of the function halves each time \(x\) increases by 1.
• For positive \(x\) values, the function approaches zero, representing a rapid decrease in value.
• For negative \(x\), values become greater than one because raising to a negative power is the same as taking the reciprocal of the base raised to a positive power.

Grasping this concept is crucial because it applies to various phenomena, including radioactive decay, depreciation of assets, and cooling processes. Observing how each increment in \(x\) halves the function's value is essential to understanding its exponential nature.

Graphing Techniques

Graphing exponential functions involves several techniques to ensure an accurate sketch. Let's explore the steps used for the function \(f(x) = \left( \frac{1}{2} \right)^{x}\).

To accurately graph:

• Identify Key Points: Choose a few specific \(x\)-values, particularly around zero, to identify notable points. For example, we use \(x=0, 1, 2, -1\) to determine their corresponding \(y\) values as 1, \(\frac{1}{2}\), \(\frac{1}{4}\), and 2 respectively.
• Plot the Points: Using a graph grid, accurately plot each of these key points. They provide a structure for your graph and help illustrate the exponential pattern.
• Draw a Smooth Curve: With points plotted, draw a curve that passes through them. The decrease should be steep as the curve moves to larger positive \(x\)-values, and it should rise sharply as it moves in the negative \(x\) direction.

Using graphing techniques, we translate abstract mathematical expressions into visual representations, helping to deepen our understanding of exponential relationships.

Asymptotic Behavior

Asymptotic behavior describes how a graph behaves as it approaches a line it never actually touches, known as an asymptote. In our function \(f(x) = \left( \frac{1}{2} \right)^{x}\), the graph exhibits particular asymptotic behavior.

Here are key points about this behavior:

• The x-axis (or \(y=0\)) is the horizontal asymptote for this function because as \(x\) increases positively, \(f(x)\) gets closer to zero but never truly reaches it.
• This demonstrates that even though the value of \(f(x)\) becomes incredibly small, it does not particularly become zero, highlighting the distinction between approaching and achieving a limit.
• For large negative \(x\), the function values rise because they take the reciprocal form, showing increasing values diverging upward.

This behavior is crucial because it shows limits and extents of exponential functions and provides insight into long-term behavior of these functions. Understanding asymptotes impacts how we interpret various exponential scenarios, from financing to population growth models.