# Slope

## Key Questions

• The slope is a number that tells you how much y changes when x changes.
For example: a slope of 5 means that for each change in x of 1 unit (for example between 6 and 7) the correspondig y changes of 5 units. This is for a positive slope, so that your value of y is getting...bigger!!!

The negative slope is the opposite, it tells you of how much y decreases for each increas of 1 unit in x.
A slope of -5 tells you that the value of y decreases of 5 units in the 1 unit interval of x: As you may guess the slope is a measure of the "inclination" of your line!!!

Try to guess what a slope of zero means!!!!!!

See below:

#### Explanation:

The steepness of a line is essentially the slope. When we see a line oriented from bottom left to upper right, it has a positive slope. A negative slope would be depicted by a line going from the upper left to bottom right.

Slope values increase the higher the number is: For instance, a line with a slope of $2$ is steeper than a line with a slope of $\frac{1}{2}$.

Hope this helps!

• Use the slope formula ($m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$) to calculate the slope given two points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$.

Here is an example of finding the slope, given two points (-2,3) and (4,-5).

$\left(- 2 , 3\right) = \left({x}_{1} , {y}_{1}\right)$
$\left(4 , - 5\right) = \left({x}_{2} , {y}_{2}\right)$

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

$m = \frac{- 5 - 3}{4 - \left(- 2\right)}$

$m = \frac{- 5 - 3}{4 + 2}$

$m = \frac{- 8}{6}$

$m = - \frac{4}{3}$

The slope of of (-2,3) and (-4,5) is $- \frac{4}{3}$

• Nuzhat has already discussed how you can find the slope of a line from two points that lie on the line. I'll discuss two other methods of finding the slope from a graph.

1. From the angle made with the x-axis

Since the slope of a line is basically the ratio of the y-component of the line to its x-component,

The slope of a line can be found out by taking tangent of the angle between the given line and the x-axis.

Consider the following figure: In this case, the angle between the x-axis and the line is $\theta$.

Therefore,
Slope of the given line = $\tan \theta$

Note: Angles in the counterclockwise direction are taken as positive, and those in the clockwise direction are taken as negative.

For example, if the angle between the x-axis and the given line is ${30}^{o}$,

Slope of the given line = $\tan 30 = \frac{1}{\sqrt{3}}$

2. From the equation of the line

The slope of a line can also be determined from its equation. The standard form of the equation of a line is:

$A {x}^{2} + B y + C = 0$

where $A , B \mathmr{and} C$ are some constants.

First, the equation of the line must be written in the standard form.

Then, the slope of the line = $- \frac{A}{B}$

For example, let the equation of the given line be ${x}^{2} + 3 = 2 y$.

Rewriting in the standard form, we get: ${x}^{2} - 2 y + 3 = 0$
and we can see that:
$A = 1$
$B = - 2$
$C = 3$

Therefore, the slope of the line $= - \frac{A}{B} = - \frac{1}{- 2} = \frac{1}{2}$

Slope is the change in the y values divided by the change in the x values

#### Explanation:

$\text{slope"="rate of change in y"/"rate of change in x" ="rise"/"run}$

$\left(\textcolor{b l u e}{{x}_{1}} , \textcolor{b l u e}{{y}_{1}}\right)$

$\left(\textcolor{red}{{x}_{2}} , \textcolor{red}{{y}_{2}}\right)$

$\textcolor{g r e e n}{m} = \frac{\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}}{\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}}$

Is is often expressed as rise over run.