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. $$ \begin{array}{|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 4 & 7 & 10 & 13 \\ \hline \boldsymbol{y} & 3.3 & 10.1 & 30.6 & 92.7 & 280.9 \\ \hline \end{array} $$

Short answer

The scatter plot does confirm an exponential relationship between \(x\) and \(y\) meaning an exponential model does fit the data. The final model might look like: \(y=a \cdot e^{b \cdot x}\), where \(a\) and \(b\) are calculated constants. The exact values of \(a\) and \(b\) will depend on the specific calculations.

Step-by-step solution

Calculate the Natural Logarithm of y

First, calculate the natural logarithm of each value of \(y\) in the given data set. Remember that the natural logarithm of a number \(k\), denoted by \(\ln k\), is equal to the power \(q\) to which the base \(e\) (approximately equal to 2.71828) must be raised to produce the number \(k\). Calculate \(\ln y\) for each \(y\) value in the data set.

Create a Scatterplot

Now, using the values calculated in the first step along with corresponding \(x\) values, create a scatter plot. Plot each point \((x, \ln y)\), where \(x\) is the \(x\)-coordinate and \(\ln y\) is the y-coordinate.

Fit a Line to the Data

See if the points on the scatter plot closely resemble a straight line. If it does, then it indicates that an exponential model probably fits the data. In such case, use the least squares method or any suitable method to fit a straight line to the data.

Find the Exponential Model

Observe the relationship between \(x\) and \(\ln y\) to derive an exponential model of the form \(y=a \cdot e^{b \cdot x}\), where \(a\) and \(b\) are constants. The value of \(a\) is the y-intercept of the line (which is \(\ln y\)'s value when \(x=0\)) and \(b\) is the slope of the line. After calculating the values of \(a\) and \(b\), retransform the y values from \(\ln y\) to \(y\) by using the exponential function, \(y=e^{\ln y}\), to find the final exponential model.

Key concepts

Scatter Plot

In exploring relationships between two sets of data, the scatter plot is an indispensable tool. It allows you to visually depict the values from a dataset and observe potential correlations or patterns. To create a scatter plot, you select a horizontal axis (usually for the independent variable, such as time or a specific condition) and a vertical axis (for the dependent variable, such as the observed outcome), and then plot points that represent data pairs from both sets.For example, in an exercise where we're interested in examining whether an exponential model fits a given set of data, the first step would usually involve plotting the given pairs of values. Each pair consists of an input value, often labeled as 'x', and an output value, in this case, the natural logarithm of 'y' (designated as \(\ln y\)). The resulting scatter plot provides a quick visual assessment of the data's behavior and is a preliminary step before delving into more complex modeling techniques.

If the scatter plot reveals that the points approximate a straight line, this suggests an underlying exponential relationship between 'x' and 'y'. This is because the exponential function, when transformed into a logarithmic scale, will exhibit linear characteristics, allowing for linear regression techniques to be applied in the next steps.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is one of the most pivotal concepts in mathematics, particularly when dealing with exponential data. It is defined as the time required to reach a certain level of continuous growth. The base of a natural logarithm is \(e\), an irrational and transcendental number approximately equal to 2.71828. In contexts where data exhibit exponential growth or decay, taking the natural logarithm of the dataset can linearize the data, allowing for simpler analysis.When you take the natural logarithm of an exponential expression like \(y = a \cdot e^{bx}\), you obtain a linear relationship of the form \(\ln y = \ln a + bx\). This is precisely why, in the provided exercise, we calculate \(\ln y\) for each \(y\) value. Once these logarithmic values are obtained, plotting them against \(x\) in a scatter plot can help us spot a straight line, revealing the exponential connection between the variables. This step is crucial for further analysis using methods such as the least squares method to fit an exponential model to the data.

Least Squares Method

The least squares method is a foundational statistical technique used to determine the best-fitting curve to a set of data points by minimizing the sum of the squares of the offsets (or 'residuals') of the points from the curve. In other words, it tries to find the line (or curve) that best represents the observed data. For linear relationships, which is often the case after taking the natural logarithm of an exponential data set, this involves finding the line that minimizes the vertical distances of all the points from the line itself.In practice, this is achieved by calculating the slope and y-intercept that define the best-fit line. The slope of this line corresponds to the rate of change between the variables, while the y-intercept represents the starting value when the independent variable is zero. In an exponential model fitting scenario, after plotting \(\ln y\) against \(x\) and determining the line of best fit, you'll derive constants that will later be used to reconstruct the exponential model, transforming the linear equation back to its original exponential form.

Exponential Equation

An exponential equation is a mathematical expression where a constant base is raised to a variable exponent. This form of equation is used to model phenomena with continuous growth or decay rates, such as populations, radioactive decay, and interest rates. The general form of an exponential equation is \(y = a \cdot e^{bx}\), where \(a\) is the initial value, \(b\) indicates the growth rate, and \(e\) is Euler's number.When analyzing data with an exponential trend, the transformation of data using the natural logarithm is vital. In the context of the exercise, we can take an equation derived from the least squares method, where the slope and intercept are determined from our scatter plot of \(\ln y\) versus \(x\). The \(a\) and \(b\) values from the equation \(\ln y = \ln a + bx\) can then be transformed to find the parameters of the exponential model \(y = a \cdot e^{bx}\). By recognizing and applying the rules of logarithms and exponents, we're able to translate our linear analysis back into an accurate exponential model that encapsulates the original data's behavior.