Question

Use the following data to work Problems 15 to 17 . Draw a graph that shows the relationship between the two variables \(x\) and \(y\) $$\begin{array}{rrrrrrr} x & 0 & 1 & 2 & 3 & 4 & 5 \\ y & 25 & 24 & 22 & 18 & 12 & 0 \end{array}$$ a. Is the relationship positive or negative? b. Does the slope of the relationship become steeper or flatter as the value of \(x\) increases? c. Think of some economic relationships that might be similar to this one.

Short answer

a. Negative. b. Steeper. c. Demand curve for price and quantity.

Step-by-step solution

Plot the Data Points

Plot each pair of \(x\) and \(y\) values on a Cartesian plane. The points to plot are (0, 25), (1, 24), (2, 22), (3, 18), (4, 12), and (5, 0).

Draw a Line through the Points

Once all points are plotted, connect them with a line to see the overall trend. This will help visualize the relationship between \(x\) and \(y\).

Determine the Relationship

Observe the line drawn in Step 2. If the line goes down from left to right, the relationship is negative; if it goes up, the relationship is positive. In this case, as \(x\) increases, \(y\) decreases, indicating a negative relationship.

Analyze the Slope

Check if the steepness of the line changes as \(x\) increases from 0 to 5. If the line gets steeper, the rate of decrease in \(y\) increases as \(x\) increases. In this case, the slope appears to become steeper as \(x\) increases.

Think of Similar Economic Relationships

Consider other situations where an increase in one variable results in a decrease in another. For instance, this is similar to a demand curve in economics where as price (x) increases, the quantity demanded (y) decreases.

Key concepts

negative correlation

In the context of economic graphs, a negative correlation between two variables means that as one variable increases, the other decreases. This is typically visualized on a graph where the line slopes downward from left to right.
For example, in the provided data set, as the value of \(x\) increases, the value of \(y\) decreases. This inverse relationship is clear by plotting the points: (0, 25), (1, 24), (2, 22), (3, 18), (4, 12), and (5, 0) and drawing a line through them. The negative slope of the line confirms a negative correlation.

slope analysis

Slope analysis involves examining the angle and steepness of the line on a graph. The slope can show how quickly one variable changes in response to changes in another variable.
Mathematically, the slope is calculated as the change in \(y\) divided by the change in \(x\) (rise over run). In our example, as \(x\) increases from 0 to 5, \(y\) decreases. Initially, the decline is gradual, but it becomes steeper as \(x\) values grow, showing an increased rate of decrease in \(y\).
Therefore, the slope becomes steeper as \(x\) increases, indicating that the impact on \(y\) becomes more significant with each unit increase in \(x\).

demand curve

A demand curve graphically represents the relationship between the price of a good and the quantity demanded. It typically shows a negative correlation because as the price (x) increases, the quantity demanded (y) decreases.
This concept is reflected in our example where the line becomes steeper as \(x\) increases. The demand curve slopes downward, illustrating that higher prices lead to lower quantities demanded.
In economics, understanding the shape and slope of the demand curve is critical. It helps in making decisions about pricing, production levels, and understanding consumer behavior.