Slope

Good afternoon, my name is Professor Maria and I am here to teach you about slope. A slope is the same thing as the coefficient on the x term. In math, the slope is the number that describes both the direction and the steepness of a line. The slope can tell us if the direction of a line is decreasing, increasing, horizontal or vertical.
A few things to note:

• In order to determine if the slope is increasing, the line must be going up from left to right. This means the slope is positive.
• A decreasing slope goes down from left to right. This means the slope is negative. 
• If a line contains a slope of zero, it means the line is horizontal. 

Now that we know what slope is, I am going to show you how to find slope and why it is important in mathematics. 

• To find the slope, you need to first pick two points on your line - you will be using them to divide the difference of the y-coordiante of a point by the difference of the x-coordiante. 

• Using the equation y2 - y1 / x2 - x1 we are able to find the slope of the line. The number on the top part of the faction is how many points on the line you are going to rise and the number on the bottom of the fraction is how many points you are going to run, in other words, move left or right depending if it is negative or positive. (If the number is positive you are going to move up or to the right and if it is negative you are going to move down or to the left.)

Now that we know how to find slope, lets use it in terms of linear functions. A linear equation is the equation of any straight line -  its equation is represented in terms of y = mx + b (this equation is called the slope-intercept form)

• In slope-intercept "m" is the slope and "b" is the y-intercept.
• In order to find the slope using the slope-intercept form you pick two points on the line and replace them for the "x"and the "y" of the equation, after solving for slope you would then use that answer to plug it in for "m" and solve for "b". By doing this, you are able to find the equation of the line as well as its intercept. 

• Example 1:

Find the equation of the line that has a slope of 4 and passes through the point (-1,-6)

y = mx + b

(–6) = (4)(–1) + b

–6 = –4 + b

–2 = b

Using the points given to us and the slope, we are able to find the y-intercept. 

- What if they don't give you the slope and ask you to put the equation of a line in terms of y = mx + b?

If they don't give you the slope you can find the slope by using the points on the line. 

• Example 2:

Find the equation of the line that passes through the points (-2, 4) and (1, 2). 

Now that I have the slope and two points I can find the intercept by solving for "b" by using the information I already have. (Using points (-2, 4) and slope -2/3)

y = mx + b




4 = (– ²/₃)(–2) + b





4 = ⁴/₃ + b





4 – ⁴/₃ = b
12/3 – ⁴/₃ = b





b = 8/3




...so  y = ( – ²/₃ ) x + ⁸/₃.

Slope can also be used to find the rate of change of a certain equation. 

• Example 3:

Todd had 5 gallons of gasoline in his motorbike.  After driving 100 miles, he had 3 gallons left.  The graph at the right shows Todd's situation.

a.  Find the slope of the line.

       

b.  What does this slope tell us?
Since , we know that Todd's bike is burning .02 gallons of gasoline for every mile that he travels.   The negative value of the slope tells us that the amount of gasoline in the tank is decreasing.


c.  What is Todd's mpg?

The    tells us that Todd can drive 50 miles on one gallon of gasoline (an mpg of 50 miles per gallon).