Slope measures the steepness of a line. The bigger the slope, the steeper the line is. The smaller the slope, the flatter it gets. Slope also tells us the direction of the line - if it goes up, down, or if it's horizontal or vertical.
Slope is calculated as the ratio of the amount of vertical change to horizontal change. The easiest way to remember this is that slope is equal to "Rise over Run." For example, if the slope of a line is 2/3, that means the line goes up 2 units for every 3 units to the right. Up 2, over 3.
There are 4 different types of slope, depending on the direction of the line. If the line goes up to the right, the slope is positive. If the line goes down to the right, the slope is negative. A horizontal line has a slope of 0. A vertical line has an undefined slope. We'll look at each type in more detail next.
Positive Slope
If the y values are increasing as x increases, the line has a positive slope. If you trace the line with your finger from left to right (the same order you read a book), the line will go up to the right. Think of a situation that has positive correlation (as one variable increases, the other also increases). Suppose you were adding $10 to your bank account each week. As time goes by, the amount of money you have in the account is increasing. If you graphed this situation, the line would go up to the right to show that the amount of money is increasing as time goes by.
Negative Slope
If the y values are decreasing as x increases, the line has a negative slope. If you trace the line with your finger from left to right (the same direction that you read a book), the line will go down to the right. Think of a situation with a negative correlation (as one variable increases, the other decreases). Suppose you're taking $10 out of your bank account each week. If you graphed this, the line would go down to the right to show that the amount of money you have left is decreasing.
Slope of Zero
If the y-values are not changing as x increases, the line will have a slope of 0. Anytime the line is horizontal (flat from left to right), the slope is zero. This would indicate a situation where there isn't any change. For example, if you just left your bank account alone and didn't put money in or take money out (and no interest was added). Essentially, you're adding 0 each week. To show that there is 0 amount of change, the graph would be a flat horizontal line to show that the amount of money isn't changing.
Undefined Slope
A vertical line has an undefined slope. In this situation, the y-values are changing, but the x-value always stays the same. Think about the bank account example. It wouldn't make sense if the graph was a vertical line. That would you mean you had all sorts of different amounts in the bank at the same exact moment, which isn't possible. If you look at the definition of slope, the amount of horizontal change is in the denominator of a fraction. In math, you can't have a 0 in the denominator. It doesn't make sense to divide by 0 so we say that the slope of a vertical line is undefined. There isn't a slope for these types of lines.
Using a Graph to Find Slope
Step 1: Determine the type of slope (Positive, Negative, Zero, or Undefined). Step 2: Find the Rise and the Run. Step 3: Write Rise/Run as a simplified fraction.
Example 1 Find the slope of the line below.
Step 1: The line is going up to the right, so the slope is positive. The y-values are increasing as x increases. Step 2: Count the spaces up and over to find the rise and the run. The line goes up 2 units and over 3 units so the rise is 2 and the run is 3.
Step 3: Slope is Rise/Run. 2/3 is already a simplified fraction, so this is the final answer for the slope. Double check the sign of your answer. The slope should be positive because the line is going up to the right.
Example 2 Find the slope of the line below.
Step 1: The line is going down to the right, so the slope is negative. The y-values are decreasing as x increases. Step 2: Count the spaces down and over to find the rise and the run.
Step 3: Slope is Rise/Run. -3 divided by 1 is just -3 so we can simplify the answer. Double check the sign of your answer. The line is going down to the right, so the slope should be negative.
Be extra careful when the line goes down to the right. A common mistake is to make both the rise and the run negative. If you go from left to right, the run will always be positive.