Problem 590 - Precalculus

Question

For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places. $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline x & {0} & {0.5} & {1} & {1.5} & {2} & {3} & {4} & {5} & {6} & {7} & {8} \\ \hline f(x) & {2.2} & {2.9} & {3.9} & {4.8} & {6.4} & {9.3} & {12.3} & {15} & {16.2} & {17.3} & {17.9} \\\ \hline\end{array}$$

Step-by-Step Solution

Answer

Plot the scatter, choose the model, and perform regression to find the equation.

Step 1: Plot the Data Points

First, enter the given pairs of \( (x, f(x)) \) from the table into a graphing utility or software that allows scatter plots. Plot each pair as a point on a coordinate plane to create a scatter diagram. Ensure all points from \( x = 0 \) to \( x = 8 \) are correctly placed.

Step 2: Analyze the Scatter Plot Shape

Examine the overall shape of the distribution of plotted points. The points should show a specific trend which suggests the type of model to be used. Observe whether the points form a curve that might be best approximated by an exponential, logarithmic, or logistic model.

Step 3: Select the Appropriate Model

Based on the trend observed from the scatter plot: - If the plot appears to rise rapidly and then slow down as it curves, a logistic model might be appropriate. - If rapidly increasing initially and then a gradual increase, consider an exponential model. - If the increase is steady and proportional, a logarithmic model might be more suitable. Identify which model fits the scatter plot's trend.

Step 4: Perform Regression Analysis

Use the regression feature in your graphing utility to determine the equations for each potential model. Enter the data points into the utility, select the type of regression (e.g., exponential, logarithmic, logistic), and perform the analysis to get the model's equation. Round the coefficients in the equation to five decimal places.

Step 5: Interpret the Results

Evaluate the accuracy of each model by looking at the regression statistics like \( R^2 \) (the coefficient of determination) to determine which model best fits the data. Usually, a higher \( R^2 \) value indicates a better fit.

Step 6: Choose the Best Fit Equation

Based on the regression analysis, select the model that best explains the trend in the data. Provide the equation form with its coefficients rounded to five decimal places. Write down the final equation that represents the data.

Key Concepts

Exponential Model

The exponential model is commonly used when data points increase at a growing rate. Imagine you're observing a scenario where the change isn't constant but accelerates as time goes on. This is typical for modeling scenarios such as population growth or compound interest.
Exponential functions have the form \( y = a \cdot e^{bx} \), where \( a \) and \( b \) are constants, and \( e \) is the base of the natural logarithm. Here, \( a \) represents the initial amount, and \( b \) indicates how rapidly the data is increasing. If you notice in your scatter diagram that data points have a steep, progressively slower climb, it's clueing you in on an exponential relationship.
Key Characteristics:

• Rapid initial increase
• Subsequent growth that steadily decelerates
• Curve never truly levels out

When performing regression analysis, choose the exponential regression option. This will compute the equation fitting the exponential trend, helping you predict future data points based on the observed pattern.

Logarithmic Model

Logarithmic models come into play when we encounter scenarios where the rate of increase decreases over time. Unlike the exponential model, where growth escalates, logarithmic models are characterized by initially rapid changes that taper off.
The form of a logarithmic function is \( y = a + b \cdot \ln(x) \), where \( \ln(x) \) indicates the natural logarithm of \( x \). The coefficient \( a \) represents the y-intercept at \( x=1 \), and \( b \) signifies the rate of growth or decay.
Identifying a Logarithmic Pattern:

• Starts with rapid change, slowing over time
• Smooth curve with diminishing returns
• No specific asymptotic boundary point

This model is often suitable for phenomena like learning curves, where initial quick progress is made that levels as more effort provides diminishing returns. Use the logarithmic regression analysis to fit your data accurately through this model.

Logistic Model

The logistic model is ideal when your data grows rapidly at first but then levels off after reaching a certain saturating point. This model is particularly useful in scenarios like population dynamics where growth is limited by resources.
It typically conforms to the function \( y = \frac{c}{1 + ae^{-bx}} \), with \( c \) representing the upper boundary or maximum capacity that cannot be exceeded. The \( a \) and \( b \) values refine the curve’s steepness and asymptotic nature.
Recognizing a Logistic Trend:

• Initial exponential-like growth phase
• Clear point of inflection where growth slows
• Levels off forming an S-shape curve

Logistic regression is particularly valuable when modeling scenarios with constraints. By applying logistic regression analysis, you get a more nuanced view of your data, distinguishing it from exponential trends where no upper limit is present.

Regression Analysis

Regression analysis is a statistical method used to identify relationships between variables. In this context, it's the tool we use to find equations that describe patterns within scatter plot data. Whether you suspect an exponential, logarithmic, or logistic trend, regression analysis can give you a deep understanding of your data’s behavior.
The method involves:

• Entering your data into regression software
• Choosing from various regression types (exponential, logarithmic, logistic)
• Interpreting outputs such as coefficients and \( R^2 \) values

Understanding \( R^2 \): The coefficient of determination, \( R^2 \), explains the proportion of the variance in the dependent variable that is predictable from the independent variable. A higher \( R^2 \) value indicates a stronger correlation at play, meaning the model fits your data well.
When conducting regression analysis, you’ll produce equations that represent data trends, allowing predictions and deeper insights into the forces shaping your data’s pattern. Whether your data exhibits exponential rise, logistic saturation, or steady logarithmic growth, regression analysis supports precise modeling and effective prediction.