Question

Create a scatter plot of the points \((x, \ln y)\) to determine whether an exponential model fits the data. If so, find an exponential model for the data. $$ \begin{array}{|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 \\ \hline \boldsymbol{y} & 18 & 36 & 72 & 144 & 288 \\ \hline \end{array} $$

Short answer

The points \((x, \ln y)\) form an approximate straight line in the scatter plot, indicating that the relationship in the original data is exponential. The slope and intercept of this line correspond to \(\ln b\) and \(\ln a\), respectively, in the exponential model \(y = ab^x\). Once these values are determined and exponentiated to get the values of a and b, the desired exponential model can be written.

Step-by-step solution

Create the Points

Transform the given x and y values into the form \((x, \ln y)\). The new points will be \((1, \ln 18), (2, \ln 36), (3, \ln 72 ), (4, \ln 144), (5, \ln 288)\). The function \(\ln y\) is the natural logarithm of y.

Plot the Points

Plot these points on a graph. For this task, it's important to get the context of each individual point correct. If the points form an approximate straight line, it indicates an exponential relationship between the x and y values in the original data.

Determine Function Form

Since we have transformed the data points using a natural logarithm and they appear linear in the scatter plot, the relationship in the original data is likely exponential, of the form \(y = ab^x\). Our task now is to find the values of a and b.

Calculate Slope and Intercept

The exponential form \(y = ab^x\) becomes \(\ln y = \ln a + x \ln b\) after taking the natural logarithm of both sides. This is a linear relation in the transformed variables, with \(\ln a\) as the intercept and \(\ln b\) as the slope. Use the two-point formula to calculate the slope and intercept of the line formed by the points in the scatter plot.

Determine Exponential Model

Use the calculated slope and intercept to find the values of a and b, through the relationships \(\ln a =\) intercept and \(\ln b =\) slope. Then, the desired exponential model is given by \(y = ab^x\).

Key concepts

Understanding Scatter Plots in Data Modeling

Scatter plots are important tools in data analysis. They help visualize relationships between two variables. In our case, the scatter plot uses points in the form \((x, \ln y)\). By plotting these, we look for a pattern or trend among our data points.
If the points form a nearly straight line, it shows a specific relationship between the variables. Here, we transform our data using the natural logarithm, which can help identify exponential relationships. This visualization process is key to spotting trends that point towards an exponential model.
A scatter plot makes understanding complex data simple and vivid. It allows us to see patterns clearly, making it easier to evaluate mathematical models.

The Role of Natural Logarithm in Exponential Modeling

The natural logarithm, denoted by \(\ln\), is fundamental when working with exponential models. It helps in simplifying multiplicative relationships into additive ones, which are easier to handle.
In the given exercise, we use \(\ln y\) to transform our original data points \( (x, y) \) to \( (x, \ln y) \). This transformation is crucial because exponential relationships become linear when a logarithm is applied. This makes it possible to use linear analysis methods to find patterns that were previously more difficult to detect.
Hence, by using the natural logarithm, we make the complex simple and open doors to effective analysis and modeling.

Linear Transformation and Its Importance

A linear transformation is a key step when converting exponential data into a linear form. By applying a natural logarithm to the y-values, we effectively transform exponential functions into linear ones. This transformation paves the way for applying linear regression techniques.
The equations transform as such: from an exponential form \( y = ab^x \) to \( \ln y = \ln a + x \ln b \). After this transformation, we can work with straight-line equations \( y = mx + c \), where \( m \) is the slope and \( c \) is the intercept.

• The transformed data helps to use simple linear algebra tools.
• It allows estimating the parameters of an exponential model.

Thus, linear transformation is a bridge that connects the complex world of exponential growth with the simpler, linear domain.

Calculating Slope and Intercept for Exponential Models

Slope and intercept calculation is the final step after transforming data into a linear form. From our transformed equation \( \ln y = \ln a + x \ln b \), the parameters are straightforward.
- **Intercept**: In the equation, the intercept is \( \ln a \). This can be obtained by noting where the line crosses the y-axis in the transformed scatter plot.- **Slope**: The slope here is \( \ln b \). It can be calculated using the formula for slope with two points: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
Once the slope and intercept are known, converting back to the exponential function is simple:

• Set \( a = e^{\text{intercept}} \)
• Set \( b = e^{\text{slope}} \)

This forms our model: \( y = ab^x \). Calculating these values is key to forming accurate exponential models and predicting future values.