Problem 10 - Algebra and Trigonometry

Question

5–10 ? Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$ h(x)=2 e^{-0.5 x} $$

Step-by-Step Solution

Answer

An exponential decay graph starting high and dropping towards zero as \( x \) increases.

Step 1: Understand the Function

First, we analyze the given function \( h(x) = 2e^{-0.5x} \). This is an exponential decay function. The base of the exponential, \( e \), is a constant approximately equal to 2.718. The negative exponent \(-0.5x\) indicates the rate of decay.

Step 2: Create a Table of Values

To sketch the graph, we need to create a table of values by choosing different \( x \) values and calculating the corresponding \( h(x) \) values. Let's choose \( x = -2, -1, 0, 1, 2 \).

Step 3: Calculate h(x) for Each x-value

Calculate \( h(x) = 2e^{-0.5x} \) for each selected \( x \):- When \( x = -2 \), \( h(x) = 2e^{1} \approx 5.44 \)- When \( x = -1 \), \( h(x) = 2e^{0.5} \approx 3.30 \)- When \( x = 0 \), \( h(x) = 2e^{0} = 2 \)- When \( x = 1 \), \( h(x) = 2e^{-0.5} \approx 1.21 \)- When \( x = 2 \), \( h(x) = 2e^{-1} \approx 0.74 \)

Step 4: Plot the Points on the Graph

Use the calculated values to plot the points on a graph: \((-2, 5.44), (-1, 3.30), (0, 2), (1, 1.21), (2, 0.74)\). These points represent the function \( h(x) = 2e^{-0.5x} \).

Step 5: Connect the Points

Join the plotted points with a smooth curve. As \( x \) increases, the points should approach the x-axis, reflecting the exponential decay tendency of the function. The graph should decrease rapidly from left to right.

Key Concepts

Exponential Decay

Exponential decay refers to a process where a quantity decreases at a rate proportional to its current value. The hallmark of exponential decay is a rapid decline at the outset, which then slows over time. In our function, \( h(x) = 2e^{-0.5x} \), the decay factor is represented by the negative exponent \(-0.5x\). The constant \(e\) is approximately equal to 2.718 and serves as the base for natural exponential functions.
Some key points about exponential decay include:

• It results in a graph that decreases as it moves from left to right.
• The rate of decay is determined by the exponent. In this case, \(-0.5x\) means the function decreases at half its initial value over unit intervals of \(x\).
• As \(x\) approaches infinity, the function value asymptotically approaches zero but never actually reaches it.

Graphing Exponential Functions

Graphing involves visualizing the behavior of a function, and exponential functions have unique characteristics. For our function \( h(x) = 2e^{-0.5x} \), the graph reflects exponential decay.
When graphing exponential functions, remember:

• The initial point is often the y-intercept, which here is when \( x = 0 \). At \( x = 0 \), the graph intersects the y-axis at \( h(0) = 2 \).
• Exponential functions such as this one will have a horizontal asymptote at \( y = 0 \), reflecting the fact that they never actually reach zero.
• The shape of the graph is a smooth curve that moves from top left to bottom right, indicating a decreasing trend.

Understanding these features aids in accurately plotting and interpreting exponential functions.

Creating Table of Values

To graph an exponential function, creating a table of values is a fundamental step. By picking values for \( x \), you calculate the corresponding \( h(x) \) values. This preparation helps you sketch the curve on a graph accurately.
The process is simple:

• Select a range of \( x \) values. It’s beneficial to include both negative, zero, and positive values to see how the function behaves on both sides of the y-axis.
• Calculate \( h(x) \) for each chosen \( x \). For instance, with \( x=0 \), \( h(0) = 2e^{0} = 2 \).
• Compile these calculations into a table. This serves as a guide for plotting your points on the graph.

Working through these steps clearly demonstrates the rate and nature of the exponential decay in the function.

Plotting a Graph

Once you have your table of values, plotting the graph involves placing each pair of \( x \) and \( h(x) \) on a coordinate system.
Here are key points to consider when plotting:

• Start by marking each point dictated by the table of values on the graph. For example, plot points like \((-2, 5.44)\) and \((2, 0.74)\).
• Ensure consistent scaling on both axes to accurately reflect the relationship between \( x \) and \( h(x) \).
• Once all points are plotted, connect them with a smooth curve to represent the exponential decay. The curve should start high on the left and drop as it moves to the right.
• The curve should flatten out and approach zero but should not touch the x-axis, highlighting the asymptotic nature of the decay.

This methodical approach to plotting ensures the graph visually communicates the properties of the exponential function.