[MUSIC PLAYING] Let's look at our objectives for today. We'll start by reviewing the coordinate plane. We'll then look at how to find the slope of a line on a graph. Finally, we'll do some examples calculating the slopes of lines with a positive, negative, zero, and undefined slope.
Let's start by reviewing the coordinate plane. A coordinate plane has a horizontal and vertical axis. The horizontal axis is the x-axis. It has a positive side and a negative side and is centered at zero.
The vertical axis is the y-axis. It also has a positive side and a negative side and is centered at zero. The intersection of the two axes is known as the origin. Points on the plane are written as an ordered pair x, y-- where x is the x-coordinate, y is the y coordinate, and the origin is at 0, 0.
Let's look at what the slope of the line tells us. Here's an example of a straight line that extends infinitely in both directions. The steepness of a line is called its "slope." The slope of a line can be calculated using the x- and y-coordinates of any two points on the line.
To calculate the slope, we divide the change in y-coordinates by the change in x-coordinates from any two points on the line. Therefore, the formula for slope is m equals y2 minus y1 over x2 minus x1. m is the variable that is used most commonly to represent the slope of a line. The 2s and the 1s in the formula relate to the x- and y-coordinates of the two points where x1 y1 is the first point and x2, y2 is the second point used to calculate the slope of the line. In the numerator, y2 minus y1 is the difference between the two y-coordinates.
And in the denominator of the fraction, x2 minus x1 is the difference between the two x-coordinates. Another way to think about slope is the change in y over the change in x or rise over run because rise describes the change in y, a vertical change, and run describes the change in x, a horizontal change. Slope is useful in many everyday situations, including price and cost, transportation fares, and inclines, such as roof tops, ski slopes, and parking ramps.
So let's do an example of finding the slope of a line. Suppose you have a landscaping business. The line below represents the relationship between time in hours and the number of lawn mowed.
We can pick any two points on the line to calculate the slope. Let's choose 2, 1 and 8, 4. We can label our points x1 and y1 and x2, y2.
Substituting these values into the formula gives us 4 minus 1 over 8 minus 2. This simplifies to 3 over 6 or 3 divided by 6, which is 1/2. So 1/2 is the slope of the line between any two points on this graph. It also means that one lawn takes two hours to mow.
Here's our second example. Suppose the temperature is dropping throughout the day. The line below represents the relationship between time in hours after 8:00 a.m. and the temperature.
By looking at the line, we can see that the slope will be negative because the line goes down as you read the graph from left to right. To calculate the slope, we pick two points-- 0, 7 and 7, 0. We can label our points x1, y1 and x2, y2.
Substituting these values into the formula gives us 0 minus 7 over 7 minus 0. This simplifies to negative 7 over 7 or negative 7 divided by 7, which is negative 1. So negative 1 is the slope of the line between any two points on this graph. This also means that the temperature is decreasing by 1 degree each hour after 8 a.m.
Here is your third example. The graph below shows the height of a teenager in feet in relation to their age in years after 18. By looking at the line, we can see that the line has a 0 slope-- meaning no steepness-- because it's a horizontal line, which means there is zero change in the values of the y-coordinates. All horizontal lines have a slope of zero because there is no change in the y-coordinates between any two points on the line.
To calculate the slope, we can pick the points 2, 6 and 6, 6. We label our points x1 and y1, x2 and y2. Substituting these values into the formula gives us 6 minus 6 over 6 minus 2.
This simplifies to 0 over 4 or 0 divided by 4, which is 0. So 0 is the slope of the line between any two points on the graph because there is no change in the y-coordinates between any two points. This also means that the person's height is not changing over time after age 18.
Here is our last example. On a very steep part of a cliff, the vertical movement as it relates to the horizontal movement of a climber is shown below. By looking at the line, we can see that the line has an undefined slope-- meaning infinite steepness because it's a vertical line, which means there is zero change in the values of the x-coordinates.
To calculate the slope, we pick the points 1, 1 and 1, 2. We label our points x1, y1 and x2, y2. Substituting these values into the formula gives us 2, minus 1 over 1 minus 1, which simplifies to 1 over 0, which is undefined because we cannot have 0 in the denominator of a fraction because we cannot divide by zero. So the slope is undefined between any two points on the graph because there's zero or no change in the x-coordinates between any two points. All vertical lines have slopes which are undefined because there will be division by zero in the slope formula.
Let's go over our key points from today. Make sure you get these in your notes so you can refer to them later. A coordinate plane has a horizontal x-axis and a vertical y-axis. The steepness of a line is called its "slope."
The slope of a line is the same between any two points on that line. Another way to think about slope is the change in y over the change in x or rise over run. And a line that is perfectly horizontal will have a zero slope while a line that is perfectly vertical will have an undefined slope.
So I hope that these key points and examples helped you understand a little bit more about finding the slope of a line on a graph. Keep using your notes, and keep on practicing. And soon, you'll be a pro. Thanks for watching.
