ঢাল নির্ণয়: গ্রাফের সাহায্যে

Finding the slope of a line on a graph is pretty similar to using points or coordinates to find a slope, but it can be a lot more visual. In this explanation, I'll walk you through the whole process.

Finding the Slope of a Line on a Graph Using Rise Over Run

If you need to find the slope of a line using a graph, first you'll need a graph. Let's use this one.

Awesome! That point is $(3,2)$ because it is $3$ steps to the right of the origin point $(0,0)$ and $2$ steps up. Now, to find the slope we need to compare two points on the line, so let's find another point:

I've chosen point $(5,6)$ as point $B$. If points aren't already chosen for you on the graph, remember that you can always just choose them yourself. Any two places on the line will do!

Next, we can use those two points to find the slope of the graph by looking at the rise and run between points $A$ and $B$, shown on this next image:

Rise is how much higher on the vertical ($y$) axis the point on the right is, compared to the point on the left. Run is how much further along the horizontal ($x$) axis the point on the right is, compared to the point on the left.

Great! The rise from $A$ to $B$ is $4$ and the run is $2$. Now, we can compare rise and run to find the slope: $$\frac{\text{rise}}{\text{run}} = \text{slope}$$

In other words, if you put rise over run, you'll get a fraction that describes the slope of your line.

Yep, it's $\frac{4}{2}$. Congrats, you have your slope! But wait, this isn't quite the last step for us. You can still simplify your slope by simplifying the fraction.

Alright, now you've got your final, simplified slope. We're done!

Overview

We've worked through a problem, but here are the steps so you can remind yourself of them later:

1. Choose any two points on a line.
2. Find the rise between those points by counting how many steps higher (or lower) the one on the right is than the one on the left.
3. Find the run between those points by counting how many more steps to the right one is than the other.
4. Place rise over run to get your slope: $\frac{\text{rise}}{\text{run}} = \text{slope}$
5. If you can, simplify your slope by simplifying the fraction.
6. Done!

Another way to find slope is to use points!

To find the slope of a line using a graph, you must first find two points and figure out their coordinates. For example:

We see the $x$- and $y$-coordinates are for $A = (4,5)$ and $B = (2,-1)$. We now use Rise over Run to find the slope "$m$".

The formula is $m=\frac{∆y}{∆x}=\frac{y_2-y_1}{x_2-x_1}$. Which means we must first find $\Delta x$ and $\Delta y$:

$$\Delta x = 4-2 = 2\quad\text{and}\quad \Delta y = 5 - (-1) = 5+1 = 6$$

So now we have the slope as $m=\frac{6}{2}=\frac{3}{1} = 3$

Practice

Use the graph above to answer the following questions

(Video) Slope from a Graph

by Friendly Math 101

This video by Friendly Math 101 goes over slope and how to find it given a line.

$\large{\text{Summary}}$

First, slope is the measure of a line's steepness or its rate of change. It is defined by the rise over the run. $$\frac{\text{rise}}{\text{run}}$$ Rise is how much you go up or down and run is how much you go to the left or right.

The first graph she looks at is

When calculating the slope given a graph, we want to choose two points on the graph. It doesn't matter which two.

Now to figure out the rise we start at the leftmost point and count how many spaces we go up or down in order to get to the next point.

As you see we need to move down $8$ units to get to the next point. So our rise is $-8$.

Next we need to figure out the run. Starting from the left point we count how my units to the left or right the next point is.

As you see we need to move to the right $2$ units to get to the next point. So our run is $+2$. Once we have both these values we can determine our slope. $$m=\frac{-8}{2}=-4$$ We use the variable $m$ to reference slope.

Try it Yourself

Special Slopes

There are two special cases for slope. When we have a completely horizontal line the slope is always $0$.

When we have a completely vertical line we say the slope is undefined.

Use a Graph to Find Slope

Sometimes, we may not be given coordinate points when trying to find the slope of a line. We can still use the graph to find the slope, though! We can first find the rise, then find the run.

For many graphs, this method is easier because you don't need to know the exact points. You can count or approximate the number of squares to count the rise and run.