两点间的斜率

Today I will be explaining to you how to find the slope or rate of change between two points on the Cartesian plane (a standard 2-dimensional graph).

Example 1

Billy the bug is about to crawl from Point A to Point B on a graph. If the coordinates of Point A are $(2,1)$ and the coordinates of Point B are $(3,3)$. What is the slope of the line Billy will crawl on to get to his desired destination of $(3,3)$?

We can start this off by finding $\frac{\text{rise}}{\text{run}} = \text{slope}$. In functional algebra, slope is defined as steepness, which we can measure by $\frac{\text{rise}}{\text{run}}$. $\frac{\text{rise}}{\text{run}}$ translates to the change in $y$ over the change in $x$ between two points.

In this problem, our two points are $(3,3)$ and $(2,1)$. To find the change in $y$ we can subtract the $y$ value of one set off coordinates from the $y$ value of the other set:

$$3-1 = 2$$.

To get the change in $x$ we do the same thing, subtracting one value from the other in the same order we used for $y$:

$$3-2 = 1$$

As mentioned earlier, we put these together as $\frac{\text{change in } y}{\text{change in } x}$ or $\frac{\text{rise}}{\text{run}}$. So, using what we just figured out, we can write our rate of change as $\frac{2}{1}$ (or $2$) and observe that this is the slope of the desired line.

Example 2

In this example, the points are farther apart but we can still use the same techniques to solve the problem. Try doing some of the steps yourself!

Billy is trying to crawl from Point A to Point B on a graph. If the coordinates of Point A are $(4,5)$ and the coordinates of Point B are $(9,13)$. What is the slope of the line Billy the bug will have to follow to get to his desired destination of $(9,13)$?

Just like before, we need to figure out $\frac{\text{change in } y}{\text{change in }x}$.

If we're subtracting $(4, 5)$ from $(9, 13)$, what is our change in $y$?

$$y_2 - y_1 = 13 - 5 = 8$$

So, our change in $y$ is $8$.

Next, if we're subtracting the points the same way, what is our change in $x$?

$$x_2 - x_1 = 9 - 4 = 5$$

So, our change in $x$ is $5$. Now, we need to turn these numbers into a fraction that will tell us our actual slope.

That's right, the slope is $\frac{8}{5}$, since the points moved (rose) $8$ points on the $y$-axis and moved (ran) $5$ points on the $x$-axis. Done!

The slope between two points is the $\frac{rise}{run}$ of the points. In other words, we find the distance between the $y$-coordinates of two points (since that's how much it's risen), and divide that by the distance between the $x$-coordinates (since that's how much it's run).

So, now that we know that, what's the slope between two points?

The slope is... $\frac{y_2-y_1}{x_2-x_1}$

To find the slope of a graphed line, all all you need are two points; let's call them $(x_1,y_1)$ and $(x_2,y_2)$.

To find the slope, just use this formula: $\frac{y_2-y_1}{x_2-x_1}$

That's all there is to it! :)

Here is an interpretation for the formula. You can think of this as a **rate of change in ** $y$. For example, if you have a velocity $6$ $ms^{-1}$ at $3s$ and $18$ $ms^{-1}$ at $7s$, you went from $6$ to $18$ in $7-3=4$ seconds. That is, you changed your velocity $12$ $ms^{-1}$ in $4$ seconds. So, your acceleration is $\frac{12}4=3$ $ms^{-1}$. Similarly, at point $(x_1,y_1)$ we have value of $y =y_1$ at $x=x_1$. And for point $(x_2,y_2)$ we have $y=y_2$ for $x=x_2$. Thus, the change in $y$ is $y_1-y_2$ for a change of $x_1-x_2$ in $x$. So, the rate of change in $y$, or change of $y$ per unit in $x$ is $\frac{y_2-y_1}{x_2-x_1}$. This interpretation has various meaning. For an instance, a line has a constant slope. That means, it changes uniformly with respect to $x$, which you can compare with uniform acceleration in physics. Moreover, this interpretation led to the basics of calculus. Anyway, hope you understand the motivation behind considering slopes of lines (or any functions).

To help you remember the formula, think of slope as "rise over run." The slope of a line is how many units you move up for each unit you move right if you are travelling along a line.

Great!