Problem 39 - Intermediate Algebra

Question

For Problems \(35-52\), graph each exponential function. $$ f(x)=\left(\frac{1}{3}\right)^{x} $$

Step-by-Step Solution

Answer

Graph the points and draw a curve showing exponential decay.

Step 1: Understanding Exponential Functions

The given function is an exponential function of the form \(f(x) = a^x\), where \(a = \frac{1}{3}\). In this context, the base value of \(a\) is between 0 and 1, which indicates that the function represents exponential decay.

Step 2: Make a Table of Values

To graph \(f(x) = \left(\frac{1}{3}\right)^x\), choose a range of \(x\)-values, such as -2, -1, 0, 1, and 2. Calculate the corresponding \(f(x)\) values: \(f(-2) = 9\), \(f(-1) = 3\), \(f(0) = 1\), \(f(1) = \frac{1}{3}\), \(f(2) = \frac{1}{9}\).

Step 3: Plot the Points on a Coordinate Plane

Plot the points \((-2, 9)\), \((-1, 3)\), \((0, 1)\), \((1, \frac{1}{3})\), \((2, \frac{1}{9})\) on the Cartesian plane to visualize how the function behaves. These points will show a decreasing pattern.

Step 4: Draw the Exponential Decay Curve

Connect the plotted points smoothly with a curve that passes through each of them and approaches the x-axis as x increases, but never actually touches the x-axis. The graph should show a steep descent initially and then flatten out as \(x\) becomes more positive.

Key Concepts

Exponential Decay

Exponential decay is a process where a quantity decreases over time at a rate proportional to its current value. In the given function, \(f(x) = \left( \frac{1}{3} \right)^x\), the base is \(\frac{1}{3}\), which is less than one. This indicates that for any positive increase in \(x\), the value of \(f(x)\) decreases. This if often imagined like a shrinking balloon, where each time it shrinks, it shrinks to a third of its current size.It's critical to notice that even as \(x\) continues to increase, \(f(x)\) will never reach zero. Instead, it continuously approaches the x-axis, illustrating the persistent nature of exponential decay. Understanding this concept is useful in various real-world applications, such as calculating depreciation in value or understanding radioactive decay.

Graphing Exponential Functions

Graphing exponential functions involves transforming algebraic equations into visual graphs, which provide an intuitive understanding of how these functions behave over a range of values. For the function \(f(x) = \left( \frac{1}{3} \right)^x\), let's break down the steps:

• Choose a set of \(x\) values: It's helpful to start with simple values like -2, -1, 0, 1, and 2 because they are easy to compute.
• Calculate corresponding \(f(x)\) values: For each x value, find the value of \(f(x)\). This step is crucial for determining where to place points on the graph.
• Notice the pattern: As \(x\) increases, \(f(x)\) decreases, highlighting the decay.

Once these calculations are complete, you can plot the points and connect them to form a smooth curve. The curve should start steep, gradually flattening as \(x\) becomes more positive, continuously decreasing toward but never touching or crossing the x-axis.

Coordinate Plane Plotting

Plotting on a coordinate plane is a foundation of graphing any mathematical function. It involves plotting points defined by their x (horizontal) and y (vertical) values, which in this case are your \(x\) and \(f(x)\) pairs. Start by drawing two perpendicular lines crossing at the origin, creating your x-axis (horizontal line) and y-axis (vertical line). Each unit along these axes is used to denote a standard measure, helping to visualize values clearly.When plotting points like \((-2, 9)\) or \((1, \frac{1}{3})\), locate the x-value on the horizontal axis and then move vertically to the y-coordinate. This sets the point's precise location. By connecting these points, you'll reveal the overall shape of the curve, reflecting the function's behavior over the chosen x-values.Coordinate plane plotting is a vital skill, not just for graphing exponential functions, but in various mathematical analyses, offering a concrete way to visualize abstract concepts.