Learning about Slope

My name is professor Jess, and today I am going to teach you how to find slope. Slope is something we learn early on in math, and a concept that we can’t forget. It is used in many different mathematical applications and we are going to look at some of those today.

The slope of a line measures how much the value of y changes for every so much that the value of x changes. Basically it is the calculation of a line’s steepness, by doing the (change in y-value) divided by the (change in x-value).  Lets take a look at the picture below:

On the graph above, we see two points, one labeled (x1, y1) and one labeled (x2, y2). Using these two values on the line, we find the vertical change, which is (y2-y1), and we divide that by the horizontal change, which is (x2-x1). This Rise over Run method will get us the slope.

We find slope to help us graph a linear function, to help us find the rate of change, and it is also a step in finding the equation of a line. Today we are just going to be focusing on the basics.

Now that we know what the slope is all about, let us put the concept into action.

Looking at the picture below, you can see that number one results in a slope of 3/2. This is a positive slope. In number two, we can see that the slope is negative.

*On a side note, if a line is vertical, then the slope will always be undefined, and if a line is horizontal, the slope will be 1. Also, if you are given the equation y=mx+b, the slope of this equation is m. 

If you are understanding this now, but want to make sure that you never forget the concept, watch the video below for a catchy and fun song to help you remember what slope is all about! Here are the lyrics if you want to sing along: 

Chrorus: "The difference of the y and the difference of the x

Also known as rise over run

Divide the two

And then reduce

Then you got slope, slope, slope, slope

"When I'm sittin' in math and I'm tryin' to find

How to get the the slope, the slope of a line

I think about the rise, and the run all the time

Then I think of this song, and I'm gonna be fine

1/2 slope come on

1 slope come on

2 slope come on

now that's three slopes

You think I'm a dope?

I'd gotta say nope

I am gonna find that slope!""

Now, lets get look at some slope problems that have some real world applications.

The average lifespan of American man has been looked at, and the equation representing the data is y = 0.2t + 73, where t = 0 corresponds to 1960.  Explain the meaning of the slope.

What is the slope? It is m = 0.2. This values tells me that, for every increase of 1 in my input variable t (that is, for every increase of one year), the value of my output variable y will increase by 0.2. C

What is the meaning of the slope? It means that, every year, the average lifespan of American man increased by 0.2 years, or about 2.4 months.


The equation for the speed (not the height) of an apple that is thrown straight up in the air is given by v = 128 – 32t, where v is the velocity (in feet per second) and t is the number of seconds after the apple is thrown.  What is the meaning of the slope?

What is the slope? It is m = –32. This value tells us that, for every increase by 1 in my input variable t, I get a decrease of 32 in my output variable v.
What is the meaning of the slope? It means that, every second, the speed decreases by 32 feet per second. Eventually the velocity becomes zero (when the apple reaches its peak), and then becomes negative (when gravity takes over and pulls the apple back down to the ground).


Fisherman in the Great Lakes Region have been recording the number of dead fish they see while fishing in the area. The Department of Environmental Conservation monitors the pollution index for the Great Lakes Region. The model for the number of fish deaths "y" for a given pollution index "x" is y = 9.607x + 111.958. What is the meaning of the slope?

What is the slope? It is m = 9.607. This value tells me that, for every increase by 1 in my input variable x, I get an increase of 9.607 in my output variable y.
What is the meaning of the slope? In means that, for every increase in the pollution index by one unit (say, from a pollution index of 6 to a pollution index of 7), there are nine or ten more fish deaths during the year.


Hope you enjoyed class today, and really understand all the aspects of slope! 

References:

Youtube video- https://www.youtube.com/watch?v=IyfpM-ruafo

Real world applications- http://www.purplemath.com/modules/slopyint.htm