Illustrating Complex Relationships In economics you will often see a complex set of relations represented graphically. You will use graphs to make interpretations about what is happening as variables in a relationship change.

Changes in the supply of corn

A change in one or more of the determinants of supply will cause a change in supply. An increase in supply shifts the supply curve to the right as from S1 to S2. A decrease in supply is shown graphically as a shift of the curve to the left, as from S1 to S3.

A change in the quantity supplied is caused by a change in the price of the product as is shown by a movement from one point to another--as from a to b--on a fixed supply curve.

Market Equilibrium

The market equilibrium price and quantity comes at the intersection of supply and demand curves. At a price of $3 at point C, firms willingly supply what consumers willingly demand.

When price is too low (say $2), quantity demanded exceeds quantity supplied, shortages occur, and prices are driven up to equilibrium. What occurs at a price of $4?

The skills you will learn in this book are to: Describe how changing the y-intercept of a line affects the graph of a line. Describe how changing the slope of a line affects the graph of a line. Describe what has happened to an equation after a line on a graph has shifted.

Identify the intersection of two lines on a graph. Describe what happens to the x and y coordinate values of intersecting lines after a shift in a line on the graph. Identify the Point of Tangency on a curve.

Determine whether a line is a tangent line. Calculate the slope at a point on a curve. Determine whether the slope at a point on a curve is positive, negative, zero, or infinity.

Identify maximum and minimum points on a curve. Determine whether a curve does or does not have maximum and minimum points.

Analyzing Lines on a Graph After reviewing this section you will be able to: Describe how changing the y-intercept of a line affects the graph of a line. Describe how changing the slope of a line affects the graph of a line. Describe what has happened to an equation after a line on a graph has shifted.

The Equation of a line

The slope is used to tell us how much one variable (y) changes in relation to the change in another variable (x).

The constant labeled "a" in the equation is the y-intercept. The y-intercept is the point at which the line crosses the y-axis.

Comparing Lines on a Graph By looking at this graph, we can see that the cost of our plain pizza is $7.00, and the cost per topping is our slope, 75 cents. This line has the equation of y = 7.00 + .75x.

Shift Due to Change in y-intercept In the graph at the right, line P shifts from its initial position P0 to P1. Only the y-intercept has changed. The equation for P0 is y = 7.00 + .75x, and the equation for P1 is y = 8.00 + .75x.

Shift Due to Change in Slope In the graph at the right, line P shifts from its initial position P0 to P1. Line P1 is steeper than the line P0. This means that the slope of the equation has gone up. The equation for P0 is y = 7.00 + .75x, and the equation for P1 is y = 7.00 + .x.

Identifying the Intersection of Lines After reviewing this section you will be able to: Identify the intersection of two lines on a graph. Describe what happens to the x and y coordinate values of intersecting lines after a shift in a line on the graph.

Intersection of Two Lines Many times in the study of economics we have the situation where there is more than one relationship between the x and y variables. You'll find this type of occurrence often in your study of supply and demand.

In this graph, there are two relationships between the x and y variables; one represented by the straight line AC and the other by straight line WZ.

In one case, the two lines have the same (x, y) values simultaneously. This is where the two lines RT and JK intersect or cross. The intersection occurs at point E, which has the coordinates (2, 4).

Examining The Shift of a Line In any situation where you are given a shift in a line: identify both the initial and final points of intersection, then compare the coordinates of the two.

Before the Shift This graph contains the two lines R and S, which intersect at point A (2, 3). Lines shifts to the right. What happens to the intersection of the two lines if one of the lines shifts?

After the Shift On the graph below, line S0 is our original line S. Lines S1 represents our new S after it has shifted. The new point of intersection between R and S is now point B (3, 4).

Example Compare the points A (2, 3) and B (3, 4) on this graph. The x-coordinate changed from 2 to 3. The y-coordinate changed from 3 to 4.

Nonlinear Relationships After reviewing this unit, you will be able to: Identify the Point of Tangency on a curve. Determine whether a line is a tangent line. Calculate the slope at a point on a curve. Determine whether the slope at a point on a curve is positive, negative, zero, or infinity.

Identify maximum and minimum points on a curve Determine whether a curve does or does not have maximum and minimum points.

Introduction Most relationships in economics are, unfortunately, not linear. Each unit change in the x variable will not always bring about the same change in the y variable. The graph of this relationship will be a curve instead of a straight line.

This graph shows a linear relationship between x and y.

This graph below shows a nonlinear relationship between x and y.

Determining the Slope of a Curve One of the differences between the slope of a straight line and the slope of a curve is that: the slope of a straight line is constant, while the slope of a curve changes from point to point.

To find the slope of a line you need to: Identify two points on the line. Select one to be (x1, y1) and the other to be (x2, y2). Use the equation:

From point A (0, 2) to point B (1, 2.5)

From point B (1, 2.5) to point C (2, 4)

From point C (2, 4) to point D (3, 8)

The slope of the curve changes as you move along it. For this reason, we measure the slope of a curve at just one point. For example, instead of measuring the slope as the change between any two points, we measure the slope of the curve at a single point (at A or C).

Tangent Line A tangent is a straight line that touches a curve at a single point and does not cross through it. The point where the curve and the tangent meet is called the point of tangency. Both of the figures below show a tangent line to the curve.

This curve has a tangent line to the curve with point A being the point of tangency. In this case, the slope of the tangent line is positive.

This curve has a tangent line to the curve with point A being the point of tangency. In this case, the slope of the tangent line is negative.

The line on this graph crosses the curve in two places. This line is not tangent to the curve.

The slope of a curve at a point is equal to the slope of the straight line that is tangent to the curve at that point.

Example What is the slope of the curve at point A?

The slope of the curve at point A is equal to the slope of the straight line BC. By finding the slope of the straight line BC, we have found the slope of the curve at point A. The slope at point A is 1/2, or .5. This is the slope of the curve only at point A.

Slope of a Curve: Positive, Negative, or Zero? If the line is sloping up to the right, the slope is positive (+).

If the line is sloping down to the right, the slope is negative (-).

Horizontal lines have a slope of 0.

Slope of a Curve: Positive, Negative, or Zero?

Both graphs show curves sloping upward from left to right. As with upward sloping straight lines, we can say that generally the slope of the curve is positive. While the slope will differ at each point on the curve, it will always be positive.

In the graphs above, both of the curves are downward sloping. Curves that are downward sloping also have negative slopes. We know, of course, that the slope changes from point to point on a curve, but all of the slopes along these two curves will be negative.

In general, to determine if the slope of the curve at any point is positive, negative, or zero you draw in the line of tangency at that point.

Example A, B, and C are three points on the curve. The tangent line at each of these points is different. Each tangent has a positive slope; therefore, the curve has a positive slope at points A, B, and C.

A, B, and C are three points on the curve. The tangent line at each of these points is different. Each tangent has a negative slope since it’s downward sloping; therefore, the curve has a negative slope at points A, B, and C.

In this example, our curve has a: positive slope at points A, B, and F, a negative slope at D, and at points C and E the slope of the curve is zero.

Maximum and Minimum Points of Curves In economics, we can draw interesting conclusions from points on graphs where the highest or lowest values are observed. We refer to these points as maximum and minimum points.

Maximum and minimum points on a graph are found at points where the slope of the curve is zero. A maximum point is the point on the curve with the highest y-coordinate and a slope of zero. A minimum point is the point on the curve with the lowest y-coordinate and a slope of zero.

Maximum Point Point A is at the maximum point for this curve. Point A is at the highest point on this curve. It has a greater y-coordinate value than any other point on the curve and has a slope of zero.

Minimum Point Point A is at the minimum point for this curve. Point A is at the lowest point on this curve. It has a lower y-coordinate value than any other point on the curve and has a slope of zero.

Example

The curve has a slope of zero at only two points, B and C. Point B is the maximum. At this point, the curve has a slope of zero with the largest y-coordinate. Point C is the minimum. At this point, the curve has a slope of zero with the smallest y-coordinate.

We can have curves that have no maximum and minimum points. On this curve, there is no point where the slope is equal to zero. This means, using the definition given above, the curve has no maximum or minimum points on it.