Question

analyze the differences in the outputs to determine whether the data are linear, quadratic, or neither. Explain. If linear or quadratic, write an equation that fi ts the data. $$ \begin{array}{|l|c|c|c|c|c|} \hline \begin{array}{l} \text { Price decrease } \\ \text { (dollars), } \boldsymbol{x} \end{array} & 0 & 5 & 10 & 15 & 20 \\ \hline \begin{array}{l} \text { Revenue } \\ \mathbf{( \$ 1 0 0 0 s ) ,} \boldsymbol{y} \end{array} & 470 & 630 & 690 & 650 & 510 \\ \hline \end{array} $$

Short answer

The relationship between price decrease and revenue is neither linear nor quadratic as the differences between the y-values are not consistent.

Step-by-step solution

Determine the Type of Relationship

Start by plotting the given points on a graph. Based on the shape of the graph, you can usually tell if the relationship is linear (straight line), quadratic (parabola), or neither.

Calculate the Differences between y-values

If there is uncertainty, calculate the differences between the y-values. A consistent difference indicates a linear relationship, a consistent difference of differences indicates a quadratic relationship. In our case, the differences are not consistent, implying a non-linear relationship.

Verify the type of relationship

The graph should resemble a parabolic curve pointing downwards, which further suggests a quadratic relationship. However, given the inconsistent differences in step 2, it is clear that the relationship is neither linear nor quadratic.

Key concepts

Linear Relationship

A linear relationship occurs when two variables have a constant rate of change. This means that when you plot the data on a graph, it forms a straight line. To determine if a relationship is linear, you can calculate the differences between consecutive y-values and see if these differences are consistent across the dataset. If they are, the data likely follows a linear pattern.

• Straight line on graph
• Consistent first differences

In the provided example, the revenue does not show a consistent difference in y-values, thus indicating that the relationship between price decrease and revenue is not linear.

Quadratic Relationship

A quadratic relationship is present when data points form a parabolic shape, usually opening upwards or downwards on a graph. This type of relationship suggests that the change between y-values is not constant, but the difference of these changes (or second difference) remains consistent.

• Parabolic curve on graph
• Consistent second differences

The data in the example fails to show consistent second differences, leading us to conclude that the relationship is neither quadratic.

Parabolic Curve

A parabolic curve is a type of graph produced by a quadratic function. It can look like a U-shape or an inverted U. The key characteristic of a parabolic curve is that it exhibits a symmetrical pattern about a vertical axis, called the axis of symmetry.

• Symmetrical U-shape or inverted U-shape
• Axis of symmetry

In our exercise, while the graph of revenue vs. price decrease might loosely resemble a downward opening parabola, the lack of consistent quadratic differences means we cannot confirm a true parabolic curve as dictated by a quadratic relationship.

Difference of Differences

The concept of difference of differences helps identify a quadratic relationship. This involves taking the difference between consecutive y-values to get the first differences, then calculating the differences between these first differences to find the second differences.

• First differences: changes between consecutive y-values
• Second differences: differences between first differences

When the second differences are consistent, you might have a quadratic relationship; however, in our scenario, the second differences vary, confirming the absence of a quadratic relationship.