Determining the Slope of a line from a Graph
Objective
So, what is the slope of a line?
According to Google, the “Slope of a Line is a number that measures its “steepness”, usually denoted by the letter m. It is the change in y for a unit change in x.”
Essentially that’s right. The slope of a line is the steepness of the line from the x axis (or the horizontal axis). So, if you think of the x axis like the floor, and the line like a hill, the slope is how we determine the hill’s steepness.
This explanation is strictly an analogy, however, so do not think this is all slope. Slope is other things too, but for now, just think of it as how steep the line is.
So why does the slope of a line matter?
The slope of a line is CRUCIAL to the line’s equation as well as how to predict what’s going to happen next.
If you are given just the slope of a line and 1 point resting on that line, you can find the entire line, equation, and x-y table easily.
Slope is also used in many calculations, from construction, to medicine, to music to computers. Pretty much any job out there will deal with the slope of something at one point or another.
Slope is also an easy way to get an A on an upcoming test (hint hint).
The many faces of slope
So, we write slope in a number of different ways because of how well it is used. It is still thought of (for lines) as the steepness of the line, however, there are a number of different ways to find it.
Slope can be called:
There are more, but these five are the more important ones. We are going to go into detail with numbers 1, 2, 3, and 4. 5 is important, but we’ll come back to that when we dive into what a function is.
Rise over Run
Rise over run. This is an easy way to think of how to determine the slope.
We can see this, since the slope is usually determined by how many spaces you go up a graph, followed by how many spaces you go to the right.
We call it rise because the y axis is usually up and down, so you either rise up or down.
We call is run because the x axis is usually horizontal (or flat with the ground), so you can run on a flat surface.
This helps to remember what order the slope looks like, simply because you can’t run up and down; and you can’t rise left to right.
An example would be:
CHANGE IN Y OVER CHANGE IN X
X | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
Y | 3 | 5 | 7 | 9 | 11 | 13 | 15 |
PRETTY SIMPLE, RIGHT?
Now that you have the slope, you can determine what the graph will look like and what the equation will look like.
It also works if you are only given 2 points and asked to determine the line (which happens in jobs all over the world.
QUICK REVIEW
When given a point, the coordinates are as follows: ( x , y ).
This means that the number to the left is the number that goes on the x-axis and the number to the right is the number that goes up from the x-axis (or the y axis).
So, the point ( 2, 1 ) on the graph looks like:
As you can see by the graph, we went over 2 spaces and up 1 space.
BACK TO SLOPE
Slope is the number next to X in a linear equation
Another example
The slope of a line
So, now that we know what slope is, and how to find it in an equation, the question becomes
What do we do if we’re just given a graph and asked to find the slope?
Well, it’s not as easy as finding the slope of an equation, however, it’s still not too difficult.
Finding the Slope From a Graph
So, to find the slope from a graph, we pick a point and count:
How far up/down (which is the numerator of the slope)
How far left/right (which is the denominator of the slope).
It sounds more confusing than it is, so let’s look at an example:
EXAMPLE:
So, if we’re given a graph like:
And we are asked to find the slope of this line, we know we simply need to locate a point, and find how far away the next point is away from it.
So, let’s take a look.
Well, if we start at the black point, and then count up…..
So, we can see that we needed to go up 3 units.
So, this means our rise for our slope is 3.
Now, we need to count how far to the left or right the point is to really find the slope.
So, if we start at where we left off and count to the right
THAT’S IT!
Again, it’s not as easy as finding the slope of a line from an equation
But it’s not too incredibly difficult.
Let’s look at a few more examples:
EXAMPLE 1:
So, if we’re given a graph like:
And we are asked to find the slope of this line, we know we simply need to locate a point, and find how far away the next point is away from it.
So, let’s take a look.
Well, if we start at the black point, and then count up…..
So, we can see that we needed to go up 1 unit.
So, this means our rise for our slope is 1.
Now, we need to count how far to the left or right the point is to really find the slope.
So, if we start at where we left off and count to the right
EXAMPLE 2:
So, if we’re given a graph like:
And we are asked to find the slope of this line, we know we simply need to locate a point, and find how far away the next point is away from it.
So, let’s take a look.
Well, if we start at the black point, and then count up…..
So, we can see that we needed to go up 2 units.
So, this means our rise for our slope is 2.
Now, we need to count how far to the left or right the point is to really find the slope.
So, if we start at where we left off and count to the right
EXAMPLE 3:
So, if we’re given a graph like:
And we are asked to find the slope of this line, we know we simply need to locate a point, and find how far away the next point is away from it.
So, let’s take a look.
Well, if we start at the black point, and then count up…..
So, we can see that we needed to go up 4 units.
So, this means our rise for our slope is 4.
Now, we need to count how far to the left or right the point is to really find the slope.
So, if we start at where we left off and count to the right