To find the slope of a line between two points, calculate the ratio of the change in <EQUATION> to the change in <EQUATION>. Solve the following problems. Click “Submit” when finished.

To find the slope of a line, you can choose two points on the line and calculate the ratio of the change in <EQUATION> to the change in <EQUATION>, or the rise to the run. For example, the slope of the line containing the points <EQUATION> and <EQUATION> is <EQUATION>.

Let’s find the slope of the line <EQUATION>. We first need to find some points on the line. So let’s make a table of values. When <EQUATION> is <EQUATION>, <EQUATION>. So <EQUATION>. When <EQUATION>, <EQUATION>. When <EQUATION>, <EQUATION>.

To find the slope, we need to pick two points on the line and find the ratio of the change in <EQUATION> to the change in <EQUATION> for those two points. If we pick the points <EQUATION>, the slope is <EQUATION>. So, the slope is <EQUATION>.

We could have picked two different points to find the slope. For example, we could have picked the points <EQUATION>. In this case, the slope would \n be <EQUATION> So, the slope \n is <EQUATION>. So, the slope is the same no matter which two points we pick.

Find the slope of the line <EQUATION> Click “Submit” when finished.

Do you notice any relationship between the equation of the line <EQUATION> and its slope? Use the slider to change the value of <EQUATION> and look at the graph of <EQUATION>. The slope of the line will be calculated. What is the relationship between the value of <EQUATION> and the slope of the line? Click “Done Exploring” when finished.

The line with equation <EQUATION> has slope <EQUATION>. What is the slope of the line <EQUATION>? Click “Submit” when finished.

Use the slider to change the value of <EQUATION> and notice how the line changes. Look at lines with positive and negative slopes. What do lines with positive slope look like? What do lines with negative slope look like? Click “Done Exploring” when finished.

A line with positive slope goes upward from left to right. A line with negative slope goes downward from left to right.

Use the slider to change the value of <EQUATION> and compare lines with different slopes. How does a line with slope <EQUATION> compare to a line with slope <EQUATION> What about a line with slope <EQUATION> What happens when the slope is <EQUATION> Click “Done Exploring” when finished.

The slope of the line measures its steepness. So, a line with slope <EQUATION> is steeper than a line with slope <EQUATION>, and a line with slope <EQUATION> is steeper than a line with slope <EQUATION>.

Let’s calculate the slope of a horizontal line. For example, let's use the line <EQUATION>. Any point on the line <EQUATION> has a <EQUATION>. So the graph of <EQUATION> is a horizontal line with <EQUATION>. To find the slope we can pick any two points on the line. Let’s pick the points <EQUATION>.

The slope is <EQUATION>, or <EQUATION>. <EQUATION> divided by any number is <EQUATION>, so the slope is <EQUATION>. In general, the slope of any horizontal line is <EQUATION>. This makes sense because slope measures steepness, and a horizontal line has no steepness.

Now let’s find the slope of a vertical line. Because slope measures steepness and a vertical line is very steep, its slope must be really large. Let’s calculate the slope of the line <EQUATION> and see if our guess is correct. Any point on the line <EQUATION> has an <EQUATION> coordinate of <EQUATION> So the graph is a vertical line with <EQUATION> intercept <EQUATION> To find the slope we can pick any two points on the line. Let’s pick the points <EQUATION>

The slope is <EQUATION> or <EQUATION>. But we can’t divide a number by <EQUATION>. So <EQUATION> is undefined which means that the slope of the vertical line <EQUATION> is undefined. In general, the slope of any vertical line is undefined. This means that the slope of a vertical line is so large that no real number is large enough to describe it.

Match each equation with its description. Click “Submit” when finished.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education