Problem 23 - Thinking Mathematically

Question

In Exercises 23-24, use a table of coordinates to graph each exponential function. Begin by selecting \(-2,-1,0,1\), and 2 for \(x\). Based on your graph, describe the shape of a scatter plot that can be modeled by \(f(x)=b^{x}, 0

Step-by-Step Solution

Answer

The scatter plot depicts a decreasing function, characteristic of an exponential function in the form \(f(x) = b^x\) where \(0 < b < 1\). The y-values decrease as x increases.

Step 1: Calculate y-values

Calculate the y-values for the given x-values (-2, -1, 0, 1, 2) using the provided formula \(f(x) = (\frac{1}{2})^x\). This will provide us with the coordinates (x, y) to plot on the graph. Each y-value is calculated by substituting the current x-value into the formula. For instance, for x = -2, \(f(-2) = (\frac{1}{2})^{-2} = 4\). This process is repeated for the other x-values, yielding y-values for each one.

Step 2: Plot the coordinates

Plot the coordinates obtained in the previous step on a graph. The vertical axis (y-axis) represents the calculated values, and the horizontal axis (x-axis) represents the selected x-values. After plotting all the coordinates, draw a smooth curve that best fits the plotted points.

Step 3: Describe the shape of the scatter plot

After drawing the graph, describe the shape of the scatter plot given that it can be modeled by the function \(f(x) = b^x\), where \(0 < b < 1\). In this case, the scatter plot is a decreasing graph, where the y-values decrease as x increases. This is characteristic of an exponential function where the base \(b\) is between 0 and 1 as per the given model.

Key Concepts

Coordinate Plotting

Coordinate plotting is a fundamental aspect of creating graphs that visually represent mathematical functions. It involves identifying and marking points on a graph based on their respective x (horizontal axis) and y (vertical axis) values. To plot the coordinates correctly, one must first calculate the y-values for each chosen x-value. In our example, where the function is defined as f(x) = \(\frac{1}{2}\)^x, we start by selecting specific values for x, such as -2, -1, 0, 1, and 2.

By substituting these x-values into the function, we can calculate corresponding y-values. For instance, with x = -2, f(-2) = \(\frac{1}{2}\)^{-2} = 4. These y-values are then paired with their x-values, giving us points like (-2, 4), (-1, 2), (0, 1), (1, 0.5), and (2, 0.25). Each pair forms a coordinate that we plot on the graph to visually represent the function. The goal is to place these points as accurately as possible using a grid system, creating a map that outlines the behavior of the function.

Scatter Plots

Scatter plots are graphical representations used to display and analyze the relationship between two different variables. Each point on a scatter plot corresponds to a pair of values on the vertical and horizontal axes, thus effectively displaying the function that relates these variables. In the context of graphing exponential functions, scatter plots can provide valuable insights into the function's rate of growth or decay.

When we graph the function f(x) = \(\frac{1}{2}\)^x, we're looking at a specific type of scatter plot where the relationship between x and y is defined by an exponential equation. As we plot the calculated coordinates, we should expect to see a distinct pattern as the y-values decrease as x-values increase, indicating an exponential decay because 0 < \(\frac{1}{2}\) < 1. The scatter plot will start high when x-values are negative and gradually fall toward the x-axis, leveling off as x increases, never actually reaching zero due to the nature of exponential decay.

Exponential Decay

Exponential decay is a specific kind of exponential function where the value of the function decreases over time. It is represented by the formula f(x) = b^x, where 0 < b < 1. This mathematical concept is prevalent in various fields such as physics, economics, and biology, often describing processes like radioactive decay, depreciation of assets, or the decrease in medication concentration in the bloodstream.

In our exercise, we work with the function f(x) = \(\frac{1}{2}\)^x, which is a classic example of exponential decay. The base of the exponent, 1/2, is less than 1 but greater than 0, which means that as x increases, f(x) gets smaller. It's essential to understand that the function never touches the x-axis; it only approaches it. This behavior forms a curve that steeply drops off for smaller x-values and gradually flattens as x increases. When graphing, this translates to a curve that rapidly decreases and slowly approaches zero without ever reaching it.