Problem 14 - Precalculus Mathematics for Calculus

Question

Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$h(x)=2\left(\frac{1}{4}\right)^{x}$$

Step-by-Step Solution

Answer

The function \( h(x) = 2\left(\frac{1}{4}\right)^x \) is an exponentially decreasing function.

Step 1: Understanding the Function

The given function is an exponential function of the form \( h(x) = 2 \left( \frac{1}{4} \right)^x \). This means that the base is \( \frac{1}{4} \) and the coefficient (or constant multiplier) is 2. The function decreases exponentially as \( x \) increases because the base is a fraction between 0 and 1.

Step 2: Create a Table of Values

To better understand the graph, let's choose a set of \( x \) values, substitute these values into the function, and calculate \( h(x) \). For example, you might choose \( x = -2, -1, 0, 1, 2 \).1. When \( x = -2 \), \( h(x) = 2 \left( \frac{1}{4} \right)^{-2} = 2 \times 16 = 32 \).2. When \( x = -1 \), \( h(x) = 2 \left( \frac{1}{4} \right)^{-1} = 2 \times 4 = 8 \).3. When \( x = 0 \), \( h(x) = 2 \left( \frac{1}{4} \right)^{0} = 2 \times 1 = 2 \).4. When \( x = 1 \), \( h(x) = 2 \left( \frac{1}{4} \right)^{1} = 2 \times \frac{1}{4} = \frac{1}{2} \).5. When \( x = 2 \), \( h(x) = 2 \left( \frac{1}{4} \right)^{2} = 2 \times \frac{1}{16} = \frac{1}{8} \).

Step 3: Plot the Points

Now, use the calculated \( (x, h(x)) \) pairs to plot points on the graph.- Plot the point \((-2, 32)\).- Plot the point \((-1, 8)\).- Plot the point \((0, 2)\).- Plot the point \((1, \frac{1}{2})\).- Plot the point \((2, \frac{1}{8})\).

Step 4: Sketch the Graph

Connect the plotted points with a smooth curve. The graph should show a rapid decrease as \( x \) becomes more positive, approaching but never reaching zero, showing its asymptotic behavior.

Step 5: Analyze the Graph

The graph is a decreasing exponential function. As \( x \) increases, \( h(x) \) approaches zero without ever reaching it. The y-intercept of the function is 2, as seen when \( x = 0 \). This reflects the constant multiplier in the function.

Key Concepts

Graphing Functions

Graphing exponential functions, like the one presented in this exercise, provides a visual way to understand the relationship between the function's variables. The given function is an exponential function, determined by the expression \( h(x) = 2 \left( \frac{1}{4} \right)^x \). This expression tells us how the output, \( h(x) \), changes as the input, \( x \), varies.

For exponential functions:

• The base determines the growth or decay rate. Since our base \( \left( \frac{1}{4} \right) \) is less than 1, the function demonstrates exponential decay.
• The graph of such a function often starts with a high value when \( x \) is negative and decreases as \( x \) becomes positive.
• Graphing involves connecting calculated points smoothly, showing the gradual change or trend.

Creating and observing the graph helps to intuitively grasp how the function's output decreases and how quickly or slowly it approaches vital components like axes or asymptotes.

Table of Values

Using a table of values is a structured approach to plot the points that describe the trend of a function graph. This involves selecting strategic \( x \) values and calculating their corresponding \( h(x) \) values based on the function expression:
Given function: \( h(x) = 2 \left( \frac{1}{4} \right)^x \).

• Choose values that cover a range of \( x \) (both negative and positive) to capture the behavior of the function across different domains.
• Calculate each. For example, for \( x = -2 \), \( h(x) = 32 \), and for \( x = 2 \), \( h(x) = \frac{1}{8} \).
• The results are compiled into ordered pairs \((x, h(x))\) that can be used for graphing.

This methodical process helps visualize and prepare for graphing, ensuring accuracy and a clear understanding of how each \( x \) changes \( h(x) \).

Asymptotic Behavior

The concept of asymptotic behavior is integral when discussing exponential functions. Asymptotes are lines that the graph of the function approaches but never actually reaches or intersects.

For our exponential decay function \( h(x) = 2 \left( \frac{1}{4} \right)^x \), the horizontal asymptote is the line \( y = 0 \).

• This means that as \( x \) becomes very large (positive), \( h(x) \) gets closer and closer to 0 without ever actually being 0.
• The graph seems to "hug" the x-axis, emphasizing the function's continuous decrease.
• An important note is that even as \( h(x) \) gets infinitesimally small, it remains a positive value.

Understanding the function's asymptotic behavior is crucial as it provides insight into the long-term behavior of the function beyond the plotted graph points.

Exponential Decay

Exponential decay describes the process where quantities decrease rapidly at a rate proportional to their current value. In mathematical terms, this is characterized by functions like our example \( h(x) = 2 \left( \frac{1}{4} \right)^x \). The base \( \frac{1}{4} \) being between 0 and 1 signifies decay.

• This decay indicates a rapid initial drop which slows as it progresses, continually decreasing.
• In the graph of this function, you quickly observe that as \( x \) increases from negative to positive, \( h(x) \) falls steeply.
• Applications of this concept are vast, from modeling population declines to radioactive decay.

Recognizing exponential decay is important in interpreting how real-world phenomena diminish, providing valuable predictive insights across various fields.