Problem 19 - Thinking Mathematically

Question

a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & 4 \\ \hline 1 & 1 \\ \hline 2 & 0 \\ \hline 3 & 1 \\ \hline 4 & 4 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The data are best modeled by a quadratic function.

Step 1: Create The Scatter Plot

Start by plotting the given data set in a scatter plot. The x-values will be plotted along the horizontal axis, and the y-values will be plotted along the vertical axis. The points to be plotted are (0,4), (1,1), (2,0), (3,1), and (4,4).

Step 2: Analyze The Scatter Plot

Once your scatter plot is drawn, study the distribution or spread of the plotted points. Check if the points seem to follow a certain trend or pattern. Are they very close to a straight line (which would suggest a linear function), do they form a curve that rises or falls rapidly (indicating an exponential or logarithmic function), or do they form a parabolic curve (indicating a quadratic function)?

Step 3: Identify The Function

Based on the scatter plot and its pattern, make a conclusion about the type of function that best models the data. In this case, the points form a curve that opens upwards, resembling a U-shape or a parabola. This indicates a quadratic function.

Key Concepts

Scatter Plot

A scatter plot is a type of graph commonly used in statistics to display the relationship between two variables. Each point on the plot corresponds to an individual data point from the provided dataset. In our example, drawing a scatter plot allows us to visually inspect the data points for any apparent relationship or trend.

To create a scatter plot, we assign one variable to the x-axis and the other to the y-axis, ensuring that the scale on both axes allows all data points to be comfortably represented. As we plot the data points from the exercise, we look for patterns such as clusters, gaps, or obvious trends, which will guide us in determining the nature of the relationship - be it linear, exponential, logarithmic, or quadratic.

Linear Function

A linear function models a constant rate of change and is graphically represented by a straight line. When analyzing a scatter plot, if the data points seem to align closely following a straight line with a uniform gradient, we would conclude that they are best described by a linear function of the form: \[ y = mx + b \
\] where \(m\) represents the slope of the line, reflecting the rate of change, and \(b\) represents the y-intercept, which is the point where the line crosses the y-axis. However, if the data points in the scatter plot create a curve or any other shape that's not straight, then we are dealing with a non-linear relationship.

Exponential Function

Exponential functions represent relationships where a quantity grows or decays at a rate proportional to its current value. This is typical in populations, investments, and natural phenomena. On a scatter plot, this relationship is indicated by data points that form a curve increasing (or decreasing) rapidly. The general form of an exponential function is: \[ y = a \cdot b^x \
\] where \(a\) is a constant that represents the initial value, and \(b\) is the base of the exponential function, dictating the direction and rapidity of the growth (or decay). This curve will be steeper for larger values of \(b\) and flatter for values of \(b\) closer to 1. Since the data in our exercise doesn't show this pattern, it suggests that the relationship is not exponential.

Quadratic Function

A quadratic function models a relationship where the rate of change is not constant but instead varies with the dependent variable, typically in the shape of a parabola. When we refer to a parabola in a scatter plot, we're observing data points that are aligned in a U-shaped curve, either opening upwards or downwards. The standard form of a quadratic function is: \[ y = ax^2 + bx + c \
\] Here, \(a\), \(b\), and \(c\) are constants, with \(a\) being the coefficient that determines the direction of the parabola's opening and the degree of its curvature. The data from our exercise, forming a U-shaped curve, indicates a quadratic function. As such, the changes in \(y\) values relative to \(x\) are not uniform; they instead follow the square of \(x\), which is indicative of quadratic behavior.