# Graph that shows a direct relationship between two variables is drawn

graph. ◇ Distinguish between linear and non-linear relationships and between relationships . A scatter diagram shows the relationship between two variables. This scatter diagram the unemployment and budget balance by drawing. The cost is in direct proportion to the number of tickets. We can draw a graph to show the relationship between two variables: 10 Number of tickets . direct relationship. a relationship between two variables that move in the same direction false: the graph shows a correlation between stock prices and production, but that does not necessarily mean that an increase in stock prices causes an.

Plot these points in the grid provided and label each point with the letter associated with the combination. Notice that there are breaks in both the vertical and horizontal axes of the grid. Draw a line through the points you have plotted.

Does your graph suggest a positive or a negative relationship? What is the slope between A and B? Between B and C? Between A and C? Is the relationship linear? What is the slope between D and E? Between E and F? Between D and F? Is this relationship linear? Answer to Try It! Here is the first graph. This curve, by the way, is a demand curve the next one is a supply curve. We will study demand and supply soon; you will be using these curves a great deal.

The slope between B and C and between A and C is the same. That tells us the curve is linear, which, of course, we can see—it is a straight line. Here is the supply curve. Its upward slope tells us there is a positive relationship between price per gallon and the number of gallons per week gas stations are willing to sell. The slope between D and E is 0. Because the curve is linear, the slope is the same between any two points, for example, between E and F and between D and F.

Explain how to estimate the slope at any point on a nonlinear curve. Explain how graphs without numbers can be used to understand the nature of relationships between two variables. In this section we will extend our analysis of graphs in two ways: Graphs of Nonlinear Relationships In the graphs we have examined so far, adding a unit to the independent variable on the horizontal axis always has the same effect on the dependent variable on the vertical axis. The slopes of the curves describing the relationships we have been discussing were constant; the relationships were linear.

Many relationships in economics are nonlinear. A nonlinear relationship Relationship between two variables in which the slope of the curve showing the relationship changes as the value of one of the variables changes. A nonlinear curve A curve whose slope changes as the value of one of the variables changes.

The relationship she has recorded is given in the table in Panel a of Figure The corresponding points are plotted in Panel b. Clearly, we cannot draw a straight line through these points. Instead, we shall have to draw a nonlinear curve like the one shown in Panel c. This information is plotted in Panel b. This is a nonlinear relationship; the curve connecting these points in Panel c Loaves of bread produced has a changing slope. Inspecting the curve for loaves of bread produced, we see that it is upward sloping, suggesting a positive relationship between the number of bakers and the output of bread.

But we also see that the curve becomes flatter as we travel up and to the right along it; it is nonlinear and describes a nonlinear relationship.

How can we estimate the slope of a nonlinear curve? After all, the slope of such a curve changes as we travel along it. We can deal with this problem in two ways. One is to consider two points on the curve and to compute the slope between those two points. Another is to compute the slope of the curve at a single point. When we compute the slope of a curve between two points, we are really computing the slope of a straight line drawn between those two points.

They are the slopes of the dashed-line segments shown. These dashed segments lie close to the curve, but they clearly are not on the curve. After all, the dashed segments are straight lines. When we compute the slope of a nonlinear curve between two points, we are computing the slope of a straight line between those two points. Here the lines whose slopes are computed are the dashed lines between the pairs of points.

Every point on a nonlinear curve has a different slope. To get a precise measure of the slope of such a curve, we need to consider its slope at a single point. To do that, we draw a line tangent to the curve at that point. A tangent line A straight line that touches, but does not intersect, a nonlinear curve at only one point.

The slope of a tangent line equals the slope of the curve at the point at which the tangent line touches the curve. Consider point D in Panel a of Figure We have drawn a tangent line that just touches the curve showing bread production at this point. It passes through points labeled M and N. The vertical change between these points equals loaves of bread; the horizontal change equals two bakers.

The slope of our bread production curve at point D equals the slope of the line tangent to the curve at this point. In Panel bwe have sketched lines tangent to the curve for loaves of bread produced at points B, D, and F. Notice that these tangent lines get successively flatter, suggesting again that the slope of the curve is falling as we travel up and to the right along it. In Panel athe slope of the tangent line is computed for us: Generally, we will not have the information to compute slopes of tangent lines.

We will use them as in Panel bto observe what happens to the slope of a nonlinear curve as we travel along it. We see here that the slope falls the tangent lines become flatter as the number of bakers rises. Notice that we have not been given the information we need to compute the slopes of the tangent lines that touch the curve for loaves of bread produced at points B and F.

In this text, we will not have occasion to compute the slopes of tangent lines. Either they will be given or we will use them as we did here—to see what is happening to the slopes of nonlinear curves. In the case of our curve for loaves of bread produced, the fact that the slope of the curve falls as we increase the number of bakers suggests a phenomenon that plays a central role in both microeconomic and macroeconomic analysis.

As we add workers in this case bakersoutput in this case loaves of bread rises, but by smaller and smaller amounts. Another way to describe the relationship between the number of workers and the quantity of bread produced is to say that as the number of workers increases, the output increases at a decreasing rate.

In Panel b of Figure Indeed, much of our work with graphs will not require numbers at all. We turn next to look at how we can use graphs to express ideas even when we do not have specific numbers.

Graphs Without Numbers We know that a positive relationship between two variables can be shown with an upward-sloping curve in a graph. A negative or inverse relationship can be shown with a downward-sloping curve. Some relationships are linear and some are nonlinear. We illustrate a linear relationship with a curve whose slope is constant; a nonlinear relationship is illustrated with a curve whose slope changes.

Using these basic ideas, we can illustrate hypotheses graphically even in cases in which we do not have numbers with which to locate specific points. Consider first a hypothesis suggested by recent medical research: We can show this idea graphically. Daily fruit and vegetable consumption measured, say, in grams per day is the independent variable; life expectancy measured in years is the dependent variable.

Notice the vertical intercept on the curve we have drawn; it implies that even people who eat no fruit or vegetables can expect to live at least a while! The graphs in the four panels correspond to the relationships described in the text.

Panel b illustrates another hypothesis we hear often: Here the number of cigarettes smoked per day is the independent variable; life expectancy is the dependent variable. The hypothesis suggests a negative relationship. Hence, we have a downward-sloping curve. As we saw in Figure We have drawn a curve in Panel c of Figure It is upward sloping, and its slope diminishes as employment rises.

Finally, consider a refined version of our smoking hypothesis. Suppose we assert that smoking cigarettes does reduce life expectancy and that increasing the number of cigarettes smoked per day reduces life expectancy by a larger and larger amount. Panel d shows this case. Again, our life expectancy curve slopes downward.

But now it suggests that smoking only a few cigarettes per day reduces life expectancy only a little but that life expectancy falls by more and more as the number of cigarettes smoked per day increases. We have sketched lines tangent to the curve in Panel d. The slopes of these tangent lines are negative, suggesting the negative relationship between smoking and life expectancy.

### Nonlinear Relationships and Graphs without Numbers

They also get steeper as the number of cigarettes smoked per day rises. Whether a curve is linear or nonlinear, a steeper curve is one for which the absolute value of the slope rises as the value of the variable on the horizontal axis rises.

Thus far our work has focused on graphs that show a relationship between variables. We turn finally to an examination of graphs and charts that show values of one or more variables, either over a period of time or at a single point in time. Key Takeaways The slope of a nonlinear curve changes as the value of one of the variables in the relationship shown by the curve changes. A nonlinear curve may show a positive or a negative relationship. The slope of a curve showing a nonlinear relationship may be estimated by computing the slope between two points on the curve.

The slope at any point on such a curve equals the slope of a line drawn tangent to the curve at that point.

We can illustrate hypotheses about the relationship between two variables graphically, even if we are not given numbers for the relationships. We need only draw and label the axes and then draw a curve consistent with the hypothesis. Consider the following curve drawn to show the relationship between two variables, A and B we will be using a curve like this one in the next chapter. Explain whether the relationship between the two variables is positive or negative, linear or nonlinear.

**Direct and Inverse Relationships**

Sketch two lines tangent to the curve at different points on the curve, and explain what is happening to the slope of the curve. The relationship between variable A shown on the vertical axis and variable B shown on the horizontal axis is negative. This is sometimes referred to as an inverse relationship. Variables that give a straight line with a constant slope are said to have a linear relationship. In this case, however, the relationship is nonlinear.

The slope changes all along the curve.

In this case the slope becomes steeper as we move downward to the right along the curve, as shown by the two tangent lines that have been drawn. As the quantity of B increases, the quantity of A decreases at an increasing rate.

### Appendix A: Graphs in Economics

You often see pictures representing numerical information. These pictures may take the form of graphs that show how a particular variable has changed over time, or charts that show values of a particular variable at a single point in time. We will close our introduction to graphs by looking at both ways of conveying information.

Time-Series Graphs One of the most common types of graphs used in economics is called a time-series graph. A time-series graph A graph that shows how the value of a particular variable or variables has changed over some period of time. One of the variables in a time-series graph is time itself. Time is typically placed on the horizontal axis in time-series graphs. The other axis can represent any variable whose value changes over time.

The grid with which these values are plotted is given in Panel b. Time-series graphs are often presented with the vertical axis scaled over a certain range. In this text, we will not have occasion to compute the slopes of tangent lines.

Either they will be given or we will use them as we did here—to see what is happening to the slopes of nonlinear curves.

In the case of our curve for loaves of bread produced, the fact that the slope of the curve falls as we increase the number of bakers suggests a phenomenon that plays a central role in both microeconomic and macroeconomic analysis. As we add workers in this case bakersoutput in this case loaves of bread rises, but by smaller and smaller amounts. Another way to describe the relationship between the number of workers and the quantity of bread produced is to say that as the number of workers increases, the output increases at a decreasing rate.

In Panel b of Figure Indeed, much of our work with graphs will not require numbers at all. We turn next to look at how we can use graphs to express ideas even when we do not have specific numbers. Graphs Without Numbers We know that a positive relationship between two variables can be shown with an upward-sloping curve in a graph.

A negative or inverse relationship can be shown with a downward-sloping curve. Some relationships are linear and some are nonlinear. We illustrate a linear relationship with a curve whose slope is constant; a nonlinear relationship is illustrated with a curve whose slope changes. Using these basic ideas, we can illustrate hypotheses graphically even in cases in which we do not have numbers with which to locate specific points.

Consider first a hypothesis suggested by recent medical research: We can show this idea graphically. Daily fruit and vegetable consumption measured, say, in grams per day is the independent variable; life expectancy measured in years is the dependent variable. Panel a of Figure Notice the vertical intercept on the curve we have drawn; it implies that even people who eat no fruit or vegetables can expect to live at least a while!

The graphs in the four panels correspond to the relationships described in the text. Panel b illustrates another hypothesis we hear often: Here the number of cigarettes smoked per day is the independent variable; life expectancy is the dependent variable. The hypothesis suggests a negative relationship. Hence, we have a downward-sloping curve.

As we saw in Figure We have drawn a curve in Panel c of Figure It is upward sloping, and its slope diminishes as employment rises. Finally, consider a refined version of our smoking hypothesis. Suppose we assert that smoking cigarettes does reduce life expectancy and that increasing the number of cigarettes smoked per day reduces life expectancy by a larger and larger amount.

Panel d shows this case. Again, our life expectancy curve slopes downward. But now it suggests that smoking only a few cigarettes per day reduces life expectancy only a little but that life expectancy falls by more and more as the number of cigarettes smoked per day increases. We have sketched lines tangent to the curve in Panel d. The slopes of these tangent lines are negative, suggesting the negative relationship between smoking and life expectancy.

They also get steeper as the number of cigarettes smoked per day rises. Whether a curve is linear or nonlinear, a steeper curve is one for which the absolute value of the slope rises as the value of the variable on the horizontal axis rises.

Thus far our work has focused on graphs that show a relationship between variables. We turn finally to an examination of graphs and charts that show values of one or more variables, either over a period of time or at a single point in time.

Key Takeaways The slope of a nonlinear curve changes as the value of one of the variables in the relationship shown by the curve changes. A nonlinear curve may show a positive or a negative relationship. The slope of a curve showing a nonlinear relationship may be estimated by computing the slope between two points on the curve. The slope at any point on such a curve equals the slope of a line drawn tangent to the curve at that point.

We can illustrate hypotheses about the relationship between two variables graphically, even if we are not given numbers for the relationships. We need only draw and label the axes and then draw a curve consistent with the hypothesis. Consider the following curve drawn to show the relationship between two variables, A and B we will be using a curve like this one in the next chapter. Explain whether the relationship between the two variables is positive or negative, linear or nonlinear.

Sketch two lines tangent to the curve at different points on the curve, and explain what is happening to the slope of the curve. The relationship between variable A shown on the vertical axis and variable B shown on the horizontal axis is negative. This is sometimes referred to as an inverse relationship.