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Magazine Chapter 5: Problem 10
5鈥�10 ? Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$ h(x)=2 e^{-0.5 x} $$
Short Answer
Expert verified
An exponential decay graph starting high and dropping towards zero as \( x \) increases.
Step by step solution
01
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.
02
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 \( x = -2, -1, 0, 1, 2 \).
03
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 \( x = 2 \), \( h(x) = 2e^{-1} \approx 0.74 \)
04
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 \( h(x) = 2e^{-0.5x} \).
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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:
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.
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:
The process is simple:
- Select a range of \( x \) values. It鈥檚 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.
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:
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.
