Problem 6 - Algebra and Trigonometry

Question

\(5-6\) mplete the table of values, rounded to two decimal places, and sketch a graph of the function. $$ \begin{array}{|c|c|}\hline x & {f(x)=2 e^{-0.5 x}} \\ \hline-3 & {} \\ {-2} & {} \\ {0} \\ {0} \\ {1} \\ {2} \\ {3} \\ \hline\end{array} $$

Step-by-Step Solution

Answer

The completed table shows exponential decay with decreasing x-values.

Step 1: Evaluate Function for Each x

To complete the table, we need to calculate the value of the function \( f(x) = 2e^{-0.5x} \) for each given \( x \). Let's start with \( x = -3 \) and proceed to \( x = 3 \).

Step 2: Calculate for x = -3

Substitute \( x = -3 \) into the function:\[f(-3) = 2e^{-0.5(-3)} = 2e^{1.5} \approx 2(4.48) \approx 8.97\]

Step 3: Calculate for x = -2

Substitute \( x = -2 \) into the function:\[f(-2) = 2e^{-0.5(-2)} = 2e^{1} \approx 2(2.72) \approx 5.44\]

Step 4: Calculate for x = 0

Substitute \( x = 0 \) into the function:\[f(0) = 2e^{-0.5(0)} = 2e^{0} = 2(1) = 2\]

Step 5: Calculate for x = 1

Substitute \( x = 1 \) into the function:\[f(1) = 2e^{-0.5(1)} = 2e^{-0.5} \approx 2(0.61) \approx 1.22\]

Step 6: Calculate for x = 2

Substitute \( x = 2 \) into the function:\[f(2) = 2e^{-0.5(2)} = 2e^{-1} \approx 2(0.37) \approx 0.74\]

Step 7: Calculate for x = 3

Substitute \( x = 3 \) into the function:\[f(3) = 2e^{-0.5(3)} = 2e^{-1.5} \approx 2(0.22) \approx 0.45\]

Step 8: Create Completed Table

After evaluating each \( x \), the completed table is:\[\begin{array}{|c|c|}\hlinex & f(x) \\hline-3 & 8.97 \-2 & 5.44 \0 & 2 \1 & 1.22 \2 & 0.74 \3 & 0.45 \\hline\end{array}\]

Step 9: Sketch the Graph

Using the completed table, plot the points \((-3, 8.97)\), \((-2, 5.44)\), \((0, 2)\), \((1, 1.22)\), \((2, 0.74)\), \((3, 0.45)\) on a coordinate plane. Connect the points with a smooth curve to represent the exponential decay of the function.

Key Concepts

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize how the function behaves across different values of x. By seeing the points and the overall shape they form, you'll get a clearer picture of the mathematical relationship. For our function, which is an exponential decay function, the graph reveals how rapidly the value decreases when the inputs (x) increase. To start plotting:

• Calculate points for the function using given x-values.
• Plot each point on the coordinate plane.
• Connect the dots with a smooth line to visualize the function.

Observe the general shape of this exponential decay graph. It starts high on the y-axis, swooping down as it moves to the right, never quite reaching zero. This distinct curve shows how values decrease quickly at first, then slow their decline as they approach zero. Understanding this visualization is crucial, as it reflects the exponential nature inherent in many natural processes.

Table of Values

A table of values is a very helpful tool when graphing functions because it lists both the x-values and their corresponding y-values (or f(x)) systematically. By filling out this table, we can easily see the exact coordinates to place on a graph.

For the function given, the process is straightforward:

• Use the formula for our given function, which is: \[ f(x) = 2e^{-0.5x} \]
• Substitute each x-value one-by-one into the function equation and solve.

After computing the values:

• For x = -3, f(x) is approximately 8.97.
• For x = -2, f(x) is approximately 5.44.
• For x = 0, f(x) equals exactly 2.
• Continue calculating for each x in the table until it's complete.

Using this table, you now have all the points needed to plot on the graph and can easily observe the behavior of the decay function.

Exponential Decay

Exponential decay refers to a process where a quantity decreases at a rate proportional to its current value. In this exercise, the function involves exponential decay because the exponent is negative.

The function we are working with is \[ f(x) = 2e^{-0.5x} \]. Here's what this means:

• The exponent \(-0.5x\) creates a decay factor less than one. As x increases, the value consistently gets smaller.
• This decreasing pattern happens quickly at first and slower as time goes on, creating a curve that levels off but never touches zero.

In real life, exponential decay is often seen in radioactive decay, depreciation of assets, or cooling of an object. Understanding this concept is essential for grasping many scientific and financial phenomena.