Problem 7 - College Algebra with Modeling and Visualization

Question

Make a scatterplot of the data. Then find an exponential, logarithmic, or logistic function \(f\) that best models the data. $$\begin{array}{cccccc} x & 1 & 2 & 3 & 4 & 5 \\ \hline y & 1.1 & 3.1 & 4.3 & 5.2 & 5.8 \end{array}$$

Step-by-Step Solution

Answer

Use an exponential model, calculated to fit the trend.

Step 1: Organize the Data

Begin by listing the given data points in pairs \((x, y)\). The provided data is: \((1, 1.1), (2, 3.1), (3, 4.3), (4, 5.2), (5, 5.8)\).

Step 2: Create a Scatterplot

Plot each data point on a coordinate plane, with the x-values on the horizontal axis and the y-values on the vertical axis. This visual representation will help identify the trend in the data.

Step 3: Identify the Appropriate Model

Examine the scatterplot to determine if the trend suggests exponential growth, logarithmic growth, or another identifiable pattern. The data shows a trend that can be modeled with an exponential or possibly a logistic function due to the increasing growth.

Step 4: Find the Exponential Model

Assume an exponential model of the form \(y = ab^x\). Use two data points to create a system of equations. For example, using \((x_1, y_1) = (1, 1.1)\) and \((x_2, y_2) = (2, 3.1)\), solve for constants \(a\) and \(b\).

Step 5: Solve for the Constants

Substitute the first point into the equation: \(1.1 = ab^1\). Similarly, substitute the second point: \(3.1 = ab^2\). This creates a system of equations: \[ \begin{aligned} &1.1 = ab, \ &3.1 = ab^2. \end{aligned} \] Divide the second equation by the first to find \(b\), then substitute back to find \(a\).

Step 6: Verify the Model

Once \(a\) and \(b\) are found, substitute other points into the model to verify accuracy. Adjust the model if necessary to better fit the given data points.

Step 7: Develop the Final Model

With determined constants \(a\) and \(b\), the exponential model \(y = ab^x\) is finalized based on the system's solution. This function should fit the observed data trend closely.

Key Concepts

Scatterplot

A scatterplot is an essential tool for visualizing the relationship between two variables. In a scatterplot, each point represents a pair of data values. For this dataset, the x-values (1 to 5) are plotted on the horizontal axis, and the y-values (1.1, 3.1, 4.3, 5.2, 5.8) on the vertical axis.
Creating a scatterplot allows us to see the overall pattern of the data.

• Helps in identifying trends or patterns.
• Useful for spotting outliers.

Scatterplots are valuable in decision-making, especially when choosing which mathematical model might best fit the data.

Exponential Function

An exponential function is a type of mathematical function that shows rapid increase or decrease. It is written as:

• \(y = ab^x\),

where:

• \(a\) is a constant representing the function's starting value.
• \(b\) is the base, showing the rate of growth if \(b > 1\) or decay if \(0 < b < 1\).

The exponential function is applied in this exercise after observing the exponential-like trend in the scatterplot. Using specific data points, you solve for \(a\) and \(b\) to create a model that mirrors the data closely. Exponential models are common in scenarios involving population growth, radioactive decay, and compound interest, among others.

Logarithmic Function

A logarithmic function is used to describe phenomena that grow rapidly and then level off. It is written as:

• \(y = a + b \ln(x)\),

where:

• \(a\) and \(b\) are constants.
• \(\ln(x)\) represents the natural logarithm of \(x\).

Logarithmic functions are particularly helpful in modeling processes that slow down over time. They appear frequently in nature and engineering, such as in sound intensity and pH calculations. Although not used in the finalization of the exercise model, recognizing logarithmic trends can point to processes that might saturate or plateau.

Logistic Function

A logistic function is a more complex model that represents growth that accelerates initially and then decelerates as it reaches a maximum limit. It is commonly written as:

• \(y = \frac{L}{1 + e^{-k(x-x_0)}}\)

where:

• \(L\) is the curve's maximum value.
• \(k\) determines the steepness of the curve.
• \(x_0\) is the x-value of the sigmoid's midpoint.

The scatterplot's observation might suggest a possible logistic growth, especially when data saturates or levels off at an upper limit. Logistic functions are frequently used in population studies where resources limit growth, such as carrying capacity in ecosystems. Recognizing logistic trends assists in understanding and predicting phenomena that grow rapidly at first but stabilize at critical points.