Rate of Change - Definition & Example

(Video) Slope Formula

by mathman1024

Here's a video by mathman1024 showing you how to find the slope of a line.

$\large{\text{Summary}}$

$$\text{slope}=\text{ rate of change } = m=\frac{\text{rise}}{\text{run}}=\frac{\Delta y}{\Delta x}$$

In math language, the greek letter, delta, $\Delta$, means "change".

Slope is the steepness of a line or the rate at which the line changes. We can calculate it by looking at two points and see how much the $y$-coordinates change (rise) and how much the $x$-coordinates change (run). The formula to solve for the slope is: $$m=\frac{y_2-y_1}{x_2-x_1}$$

So if we were given two points, $P(5,-7)$ and $Q(-3,3)$, what would be the slope of the line that they're on? Well we should first label our points: $$(5,-7)\text{ and }(-3,3)\\ (x_1,y_1)\text{ and }(x_2,y_2)$$ We can plug these numbers directly into our slope equation! $$m=\frac{3-(-7)}{(-3)-5}=\frac{10}{-8}$$

So if we plot our first point, $P(5,-7)$.

The rise of the slope is $10$. So let's move our point up by $10$ units.

The run of the slope is $-8$. So let's move our point to the left by $8$ units.

And now we're at our second point, $Q(3,-3)$. These are two distinct, discrete data points.

The slope of a line is the rate of change

Rate of Change: The speed at which a variable changes over a specific period of time. It is often used when speaking about momentum, and it can generally be expressed as a ratio between a change in one variable relative to a corresponding change in another. Graphically, this ratio is represented by the slope of a line.

TL;DR: The bigger the slope, the steeper the line.

You'll recognize this average rate of change from the real world: miles per hour!

Average rate of change

Sometimes, graphs won't just be one straight line and will have lots of different slopes. You can find the average rate of change between two points by finding the rise and run between them.

The average rate of change of a function $f(x)$ over an interval between two points $(a, f(a))$ and $(b, f(b))$ is the slope of the line connecting the two points: $$\frac{y_2-y_1}{x_2-x_1}=\frac{f(b)-f(a)}{b-a}$$

Let's think about this in terms of speed. If we start out driving slowly and then pick up speed as we go on, our changes over the course of our drive. However, we can still find our average speed (our average rate of change) the way we would normally find the slope between two points. Take a look at the graph below:

The function that describes our actual speed is in red. The blue line connects the two points that we need to find the average rate of change (slope of the blue line). We can notice that the red line is curving at different rates, so there is different slopes. In order to find an average slope, we want to find the slope of the blue line because of the slope of the blue line is constant.

Examples!

Example 1:

Let $f(x) = mx + b$. Find the average rate of change of $f$ between the points where $x = a$ and $x = a + h$ by plugging them into the equation for finding the secant line:

$$\frac{f(b) - f(a)}{b - a}$$

Solution:

$$\begin{align} &=\frac{f(a + h) - f(a)}{(a + h) - (a)}\\ &=\frac{(m(a + h) + b) − (m(a) + b)}{a + h - a}\\ &=\frac{(m(a) + m(h) + b − m(a) − b}{h}\\ &=\frac{m(h)}{h}\\ &=m \end{align}$$

So, the average rate of change is just the normal slope of a linear function!

Example 2:

In this case, we have set of ordered pairs as our data. Since we can calculate rate of change by just using two points, we don't actually need the full function! $$\begin{align} \text{(year, price)} = \{&\text{(1987, 3.91), (1991, 4.21), (1995, 4.35), (1999, 5.06)}\\ &\text{(2003, 6.03), (2007, 6.88), (2009, 7.50)\}} \end{align}$$ Calculate the average rate of change of ticket price with respect to time over the period $1999$ to $2009$.

Solution:

$$\frac{y_2 - y_1}{x_2 - x_1}$$ $$\begin{align} &=\frac{7.50 - 5.06}{2009 - 1999}\\ &=\frac{2.44}{10}\\ &=0.244 \end{align}$$

So, the ticket price increased an average of $\$0.24$ per year between $1999$ and $2009$.

This concept of slope will become so important as we start to work more and more with linear and non-linear functions. Spending more time with this concept and practice how to find slopes will help you a lot as we work with more functions and equations!

(Video) Slope

by Expii

This video by Expii teaches about slope.

$\large{\text{Summary}}$

What is Slope?

Slope, or rate of change, is essentially the "steepness" of a line. We refer to it as the rise over run because it's the ratio of how much something rises (moved upward or downward) and how much it runs (moves left or right). $$\text{slope}=\frac{\text{rise}}{\text{run}}$$

Slope in Math

It isn't really math unless we have numbers and expressions. Let's look at slope in a more mathematical light.

First off, we use the letter $m$ to denote slope. Second, we express the "rise" as the change in the $y$ direction and "run" as the change in the $x$ direction. $$\text{slope}=\frac{\text{rise}}{\text{run}}\quad\Longrightarrow\quad m=\frac{\text{change in }y}{\text{ change in }x}$$ The "change in" is just the difference. If we went from a height of $1\text{ ft.}$ to a height of $10\text{ ft.}$, our "change in" height is: $$10-1=9\text{ ft.}$$ Similarly, if we go from a height of $y_1\text{ ft.}$ to a height of $y_2\text{ ft.}$, our "change in" height is: $$y_2-y_1$$ Using this, we can make the mathematical equation for slope. $$\boxed{m=\frac{y_2-y_1}{x_2-x_1}}$$

Keep track of which coordinates you designate as $x_1, x_2, y_1$, and $y_2$. This is the part that most students get tripped up on. If you mix these variables up, you'll get the wrong slope.

Let's take a look at the slope of the line above, but now on a coordinate plane.

The line goes from the point $(-2,0)$ to the point $(2,2)$. We can use our new slope formula to figure out the line's slope.

When using this formula it is important to pick which point will be your $(x_1,y_1)$ and which will be your $(x_2,y_2)$. It doesn't matter which you choose for which but it's important to stick with that throughout the problem. I'm going to let: $$(x_1,y_1)=(-2,0)\\ \text{and}\\ (x_2,y_2)=(2,2)$$

Once we have this, we can begin to set up our formula. $$m=\frac{y_2-y_1}{x_2-x_1}$$ Start by putting in our $y$-points. $$m=\frac{2-0}{x_2-x_1}$$ Then, put in our $x$-points. $$m=\frac{2-0}{2-(-2)}$$ Finally, solve for $m$. $$\begin{align} m&=\frac{2-0}{2-(-2)}\\ m&=\frac{2}{4}\\ m&=\frac12 \end{align}$$

Getting to the Cafe

In the video, the girl has three options: