Problem 26 - Brief Applied Calculus

Question

$$ \text { Graph each function } $$ $$ f(x)=\left(\frac{1}{3}\right)^{x} $$

Step-by-Step Solution

Answer

Graph the function by plotting key points around the origin and sketching a curve that shows exponential decay.

Step 1: Identify the Function Type

The given function is an exponential function, which generally has the form \( f(x) = a^x \), where \( a \) is a constant. In our case, \( a = \frac{1}{3} \). This indicates an exponential decay, as the base is between 0 and 1.

Step 2: Determine Key Points

To graph the function, select a few key points by substituting values for \( x \). For instance, compute \( f(-2), f(-1), f(0), f(1), f(2) \):- \( f(-2) = \left(\frac{1}{3}\right)^{-2} = 9 \)- \( f(-1) = \left(\frac{1}{3}\right)^{-1} = 3 \)- \( f(0) = \left(\frac{1}{3}\right)^{0} = 1 \)- \( f(1) = \left(\frac{1}{3}\right)^{1} = \frac{1}{3} \)- \( f(2) = \left(\frac{1}{3}\right)^{2} = \frac{1}{9} \)

Step 3: Plot the Key Points

On a coordinate plane, plot the key points you calculated in Step 2: - \((-2, 9)\)- \((-1, 3)\)- \((0, 1)\)- \((1, \frac{1}{3})\)- \((2, \frac{1}{9})\).

Step 4: Draw the Graph

Using the plotted points, sketch the curve of the function. As \( x \) increases, the value of \( f(x) \) approaches 0 but never actually reaches zero, forming a curve that starts high on the left and approaches the x-axis as it moves right. Ensure the curve reflects the exponential decay indicated by the base.

Step 5: Verify that the Graph Matches Expectations

Check that the graph correctly shows the behavior of an exponential decay function, with the curve passing through the points plotted and asymptotically approaching the x-axis as \( x \) increases.

Key Concepts

Graphing Functions

To effectively graph an exponential function, start by understanding the nature of the function you are dealing with. The function given is \( f(x) = \left( \frac{1}{3} \right)^{x} \). This type of function often presents either exponential growth or decay.

• For exponential growth, the base \( a \) is greater than 1.
• For exponential decay, the base \( a \) is between 0 and 1.

In our example, since \( a = \frac{1}{3} \), we are observing an exponential decay. This implies that as \( x \) increases, the function value \( f(x) \) decreases, approaching zero but never becoming negative or actually reaching zero. The graph of this function will start higher on the left side and diminish towards the right, always staying positive.

Exponential Decay

Exponential decay is a phenomenon where quantities reduce at a rate proportional to their current value. In the function \( f(x) = \left( \frac{1}{3} \right)^{x} \), the base \( \frac{1}{3} \) represents this decay rate. This means:

• The function's value significantly drops as the exponent \( x \) grows.
• On a graph, this manifests as a downward slope that approaches the x-axis.
• The graph remains above the x-axis, evidencing that the values are always positive.

The quick reduction in values is typical of exponential decay, showcasing how the function shrinks rapidly but never quite hits zero. This principle is key to understanding how quickly resources deplete, populations decline, or radioactive substances decay over time.

Coordinate Plane Plotting

Plotting an exponential function on the coordinate plane involves selecting strategic points to provide an accurate representation of the curve. Start by computing values for different \( x \)-values, like in the original solution:

• Calculate values such as \( f(-2), f(-1), f(0), f(1), \) and \( f(2) \).
• Render these points on the coordinate plane: \((-2, 9)\), \((-1, 3)\), \((0, 1)\), \((1, \frac{1}{3})\), \((2, \frac{1}{9})\).

Once these points are plotted, carefully sketch the curve that connects them smoothly and follows the trend of exponential decay. The curve will be steep initially and gradually flatten as it approaches the x-axis. Remember, all y-values must be positive due to the nature of the exponential function. This method of plotting ensures clarity in visualizing how the function behaves and decays over the x-axis as you move right.