Used in everything from basic graphing to more advanced concepts like linear regression, slope is one of the primary numbers in a linear formula. It can be positive increasing upward or negative decreasing downward.
Slope is a measure of the difference in position between two points on a line. If the line is plotted on a 2-dimensional graph, the slope represents how much the line moves along the x axis and the y axis between those two points.
Though slope may appear as a whole number at times, it is technically a ratio of the x and y movement. If a given line was. A line is said to have positive slope if it increases along both the x and y axis as it moves from left to right.
If the line decreases along the y axis as it moves from left to right, it is said to have a negative slope. A line that moves horizontally or vertically without any movement along the other axis has zero slope with vertical lines sometimes being said to have infinite slope. When sketching lines on a graph, lines with positive slope move "up" when traveling left to right while those with negative slope move "down.
In this lesson, we are going to look at the "real world" meanings that the slope and the y -intercept of a line can have, in context. In other words, given a "word problem" modelling something in the real world, or an actual real-world linear model, what do the slope and intercept of the modelling equation stand for, in practical terms?
Back when we were first graphing straight lines, we saw that the slope of a given line measures how much the value of y changes for every so much that the value of x changes. For instance, consider this line:. This means that, starting at any point on this line, we can get to another point on the line by going up 3 units and then going to the right 5 units.
But and this is the useful thing we could also view this slope as a fraction over 1 ; namely:. This tells us, in practical terms, that, for every one unit that the x -variable increases that is, moves over to the right , the y -variable increases that is, goes up by three-fifths of a unit.
While this doesn't necessarily graph as easily as "three up and five over", it can be a more useful way of viewing things when we're doing word problems or considering real-world models.
Slope: Very often, linear-equation word problems deal with changes over the course of time; the equations will deal with how much something represented by the value on the vertical axis changes as time represented on the horizontal axis passes. An exercise might, say, talk about how the population grows, year on year, in a certain city, assuming that the population increases by a certain fixed amount every year.
For every year that passes that is, for every increase of 1 along the horizontal axis , the population would increase that is, move up along the vertical axis by that fixed amount.
For a time-based exercise, this will be the value when you started taking your reading or when you started tracking the time and its related changes. In the example from above, the y -intercept would be the population when the sociologists started keeping track of the population. Advisory: "When you started keeping track" is not the same as "when whatever it is that you're measuring started".
Using the example above, your population-growth model might be very accurate for the years through , but the city whose population is being measured might have been founded way back in When we talk about slope or steepness, the way it's defined is by change of Y over change of X.
Like in a fraction. Change in Y on top of change in X. That's why we call it a ratio. Remember ratio is like a fraction. So if I were to draw a little triangle here that represents how steep my line is, this would be my change in Y piece because Y is up and down. This would be my change in X piece, because X is horizontal. And whatever those numbers were on the graph, I would write as a fraction.
That's one thing you want to keep in mind. Sometimes we write it using this little triangle. This triangle is the Greek letter delta, which is tricky. Not only do you have to learn math but now you have to learn Greek. This means change in Y on top of change in X. That delta just represents the word change. And a third way we write this is using the letter M. M stands for slope, and if I had two points, I'll use them up here.
Let's say I had this point I'm going to call it X from my first point and then Y from my first point. Here's my second point. X from my second point I'm going to use that little 2 to show it's my second point. Y for my second point. Then there's a formula I could use using those X and Y numbers to find M, or the slope. This is the same thing just written in a different way. I'm finding out how much did my Y values change and putting that on top of how much did my X values change in a fraction.
So this formula is really important anytime you have two points like this. These little numbers down here are tricky. It doesn't mean take your Y value and multiply it by 2 or take your Y value and multiply it by 1.
What it means is we're just notating using what's called a subscript.
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