# Unit 1: Analyzing Lines on a Graph

## Objectives

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 equation of a straight line is given on the right. In this equation: "b" is the slope of the line, and "a" is the y-intercept, Each of these will be defined below. (NOTE: The equations of a line is frequently shown as y = m x + b.)

### Slope

 The slope is used to tell us how much one variable (y) changes in relation to the change in another variable (x). This can be written as follows:

### y-intercept

 The constant labeled "a" in the equation is what is called the y-intercept. The y-intercept is the point at which the line crosses the y-axis. To review the terms given here, you may wish to refer to the glossary.

NOTE: In this section we are not going to go into a detailed review of the equation of a line and how it is represented graphically. If you feel you need to review this, you should work through Part Two of tutorial 4, Equations and Grapahs of Straight Lines, of this series before continuing.

## Comparing Lines on a Graph

One skill that is extremely useful within economics is the ability to draw conclusions about what is going on by examining the graphs. There are many things you can tell about relationships by simply looking.

 For example, the graph here shows the relationship between the number of toppings you select to put on a pizza and the final cost of a pizza (line P). Notice the line in this graph is labeled using a letter. This is one way to label a line. 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. When comparing the lines we find: P0 to P1, we can see the only change is that the line has shifted up. Only the y-intercept, the point where the line crosses the y-axis, has changed. In other words, the initial price of the pizza has risen. 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. When comparing the lines we find: Both lines P0 and P1 still crosses the y-axis at 7, but the line goes up at a different angle. Line P1 is steeper than the line P0. This means that the slope of the equation has gone up, and, in fact, has increased to 1. The equation for P0 is y = 7.00 + .75x, and the equation for P1 is y = 7.00 + .x.

In general, when a line shifts in such a way that it maintains the same steepness as the original line, but moves up or down, or to the right or left, the y-intercept changes while the slope remained the same. If the line changes steepness, the slope must have changed.

### Example

The straight line Q in the graph below is given by the equation y = t + px. If the line shifts from its initial position at Q0 to a the new position Q1, what must have changed in the equation?

In this example the line shifted to the right but did not change its slope. Therefore only y-intercept, represented by the constant "t," changed in the equation.

If you extend both lines to intersect the y-axis, you will find Q1 intersects it at a smaller number so "t" must have decreased. (NOTE: The line Q1 intersects the y-axis below the x-axis. This means that the y-intercept of Q1 is a negative number.) The constant "p," which is the slope, remains the same.

### Example

The straight line R in the graph below is given by the equation y = Lx + Q. If the line shifts from its initial position at R0 to a the new position R1, what must have changed in the equation?

The line has changed in steepness. This means the slope, or the "L" constant changed. Since R1 is steeper, "L" increased.

You may want to make some statement about the constant "Q." In a situation where we cannot see if the two lines both intercept the y-axis in the same place, it is hard to tell if "Q" changed or not. If you do extend both lines through the y-axis, you will find they have the same y-intercept, which means "Q" does not change. Unless we do this, we cannot make any statement about "Q" but can be certain that "L" does change.

You are now ready to try a practice problem. If you have already completed the first practice problem for this unit you may wish to try the additional practice.