Problem 14 - College Algebra

Question

Enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic. $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}{x} & \hline {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} & {10} \\ \hline f(x) & {2.4} & {2.88} & {3.456} & {4.147} & {4.977} & {5.972} & {7.166} & {8.6} & {10.383} & {12.383}\\\ \hline \end{array}$$

Step-by-Step Solution

Answer

The data most closely represents an exponential function.

Step 1: Enter Data into Calculator

Input the provided values of \(x\) and the corresponding \(f(x)\) into a graphing calculator. This typically involves accessing the data or list section of the calculator and entering \(x\) values in one column and \(f(x)\) values in another.

Step 2: Generate Scatter Plot

Using the graphing calculator, create a scatter plot of the data. This involves selecting the graphing mode and ensuring the correct lists for \(x\) and \(f(x)\) are chosen to plot each data point on the graph.

Step 3: Analyze the Scatter Plot

Examine the shape of the scatter plot. Determine if the pattern of points resembles a straight line (linear), a curve that increases at an increasing rate (exponential), or a curve that gradually levels off (logarithmic).

Step 4: Test for Linearity

Check if the data fits a linear pattern. Use the calculator to find the line of best fit and check the correlation coefficient. If it is close to 1, the data may be linear.

Step 5: Test for Exponential Fit

Plot an exponential regression on the calculator if the scatter plot appears to follow an exponential growth pattern. If the model fits well and has a high coefficient of determination \(R^2\), the function is likely exponential.

Step 6: Confirm Exponential Model

After trying exponential regression, check the \(R^2\) value. A value close to 1 indicates a good fit for an exponential model. The increasing rate of \(f(x)\) suggests exponential growth.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that helps in plotting data and visualizing mathematical functions. It can be a handheld device or an app that you install on your computer or smartphone. To get started, first access the data or list section of the calculator. Here, you can input your data points—these are represented as

• the independent variable, usually denoted as \(x\), in one list
• the dependent variable, called \(f(x)\) or \(y\), in another list

Once data entry is complete, you enable the graphing mode. Then, plot each data point on a scatter plot to visually interpret the trend. This visual representation helps in analyzing whether the data aligns with a particular mathematical model, like linear, exponential, or logarithmic functions.

Exponential Function

An exponential function is a mathematical model where growth or decay happens at a consistent multiplicative rate. This is often represented in the form \(f(x) = a \, b^x\), where \(a\) is a constant and \(b\) is the base of the exponential equation. When plotting data:

• If the points on the scatter plot increase rapidly and form a smooth upward curve, it's indicative of an exponential function.
• For data exhibiting exponential growth, the value of \(f(x)\) increases at an accelerating rate as \(x\) increases.

To test if your data follows an exponential model, you can use your graphing calculator to perform an exponential regression analysis. If the resulting model has a high \(R^2\) value, close to 1, it confirms a good fit, signaling exponential growth characteristics.

Linear Function

A linear function is characterized by a constant rate of change, forming a straight line when plotted on a graph. Mathematically, it is expressed as \(f(x) = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

• When data points on a scatter plot align closely along a straight path, they are likely adhering to a linear function.
• If the rate of increase or decrease in the data is uniform, it suggests linearity.

In a graphing calculator, determining linearity involves calculating the line of best fit—a method for finding a line that best represents the data points. A high correlation coefficient close to 1 indicates the data fits a linear model well, confirming a consistent linear relationship between \(x\) and \(f(x)\).

Logarithmic Function

A logarithmic function involves a relationship where one variable increases logarithmically as another variable increases. It's generally represented by the equation \(f(x) = a \, \log_b(x) + c\), where \(a\) and \(c\) are constants, and \(b\) is the base of the logarithm.

• Logarithmic functions often describe systems that grow rapidly at first but then slow down over time, gradually leveling off.
• On a scatter plot, this is seen as a rapid ascent initially, followed by a slower, leveling increase.

To identify if your data fits a logarithmic model using a graphing calculator, attempt a logarithmic regression. Significantly, if the curve matches closely with an expected logarithmic form and has a high coefficient of determination \(R^2\), it suggests that a logarithmic function is a suitable representation of your data.