Slope Notes
*Watch these two videos first!
https://www.youtube.com/watch?v=zihsQC0IUd8
https://www.youtube.com/watch?v=6kBN9SweFI
Lesson 4.1
o What does it mean? If you are given the equation x = 4, all the points on the graph that you would draw would have a xcoordinate of 4. (4, 0); (4, 1); (4, 2); (4, 3), etc.
o If you plotted those points on the graph and drew a line through them, it would be vertical.
· In the coordinate plane, the graph of y = b is a horizontal line
o What does it mean? If you are given the equation y = 5, all the points on the graph that you would draw would have a ycoordinate of 5. (1, 5); (0, 5); (1, 5); (2, 5), etc.
o If you plotted those points on the graph and drew a line through them, it would be horizontal
· The graph of an equation involving two variables, x and y, is the collection of all points (x, y) that are solutions of the equation.
For example in the equation y = 2x + 4, plug in a zero for x and solve for y. This would give you y = 2(0) + 4. Do Order of Operations next. So your one point on the line would be (0, 4). Then plug in a 1 for x and solve for y. This would give you y = 2(1) + 4. Do Order of Operations next. So your other point would be (1, 6).
Lesson 4.2
· Slope = rise The numerator is called the “rise” = the change in y
run The denominator is called the “run” the change in x
· To find the slope when given 2 points,
m = y₂  y₁
x₂  x₁
o Step 1: Label EVERYTHING
o Step 2: Substitute numbers for letters
o Step 3: Solve
o Step 4: REDUCE! REDUCE! OR SIMPLIFY!
· What type of slope do I have? Positive, Negative, Zero, or Undefined
o If it is Positive, my line will rise from left to right
o If it is Negative, my line will fall from left to right
o If it is Zero, I will have a horizontal line (ex: y = 3)
o If it is Undefined, I will have a vertical line (ex: x = 4)
***Remember this rhyme: Undefined mean vertical line
***Another way to memorize this is UV. These letters are right next to each other in the alphabet.
Lesson 4.2 Extension
o Here are examples: y= 4x + 7 and y = 4x 10

Both of these have a slope of 4, so they would be parallel.

Lines that intersect and form right angles are perpendicular lines. Two nonvertical lines are perpendicular when their product of their slopes are 1.

For example in the equations and , the slopes of the equations are 3 and  . When you multiply them together, they equal 1.
Lesson 4.3

When two quantities x and y are proportional, the relationship can be represented by the direct variation equation , where m is the constant of proportionality. The graph of is a line with a slope of m that passes through the origin

If you are given a table and asked whether it is proportional, look to see if you a constant rate of change (slope) or you can work backwards in the table towards 0.

For example in the table below, this would be proportional:
Lesson 4.4

Slopeintercept form :
y = 2^{nd} number of the coordinate that satisfies the equation
m = the slope
x = the 1^{st} number of the coordinate that satisfies the equation
b = the yintercept

The xintercept is the point where the line crosses the xaxis

To find the xintercept of the line, plug in a 0 for y and solve for x. Here is an example:
o 2x + 3y = 6
o 2x + 3(0) = 6
o 2x = 6
o x = 3

The yintercept is the point where the line crosses the yaxis

To find the yintercept of the line, plug in a 0 for x and solve for y. Here is an example:

o 2x + 3y = 6

o 2(0) + 3y = 6

o 3y = 6

o y = 2

o Your ordered pair (0, 2). You would then plot this point on the graph.

You may also use the “cover up method” where you place your hand other the x or y term and then solve for the other letter.

Now you have the 2 points that are on the graph of the equation. After you plotted both the xintercept and yintercept., connect them with a straight line with arrows at the end of them. REMEMBER THAT ALL YOU NEED IS 2 POINTS TO CREATE A LINE. The easiest 2 points to use are the xintercept and yintercept because anything times zero equals zero!
A COUPLE OF THINGS TO REMEMBER:
Lesson 4.5
· How do I change the linear equation from standard form to slope intercept form?
o If the equation looks like this:
Ax + By = C
o Option #1:
o Let’s say you had an equation
o Your goal is to make it look like
o First, subtract 10x from both sides (inverse operation).
This would give you
o Second, divide both sides by 6 (inverse operation).
This would give you
This is the answer!
o Option #2:
o To find the slope (m) =
o To find the yintercept (b) =
o Here’s an example:
m = 2
b = 7
So your equation written in slopeintercept form is:
o Use Option #3:
o Use the xintercept and the yintercept and draw the line
Lesson 4.6
· Using a graph to write an equation of a line
o Find the yintercept from the line
o Find another point on the line (preferably the xintercept)
Calculate slope: m = y₂  y₁
x₂  x₁
o Look at the line to check if the calculated slope is the same (positive, negative, zero, or undefined)
o Write in slopeintercept form
· Writing an equation of a line that passes through two points
o Use slopeintercept form
o Substitute for x and y with the point given
o Substitute for m
o Solve for b (the yintercept)
o Now, you know the slope (which was given to you) and the yintercept
o Rewrite in slopeintercept form
o Check with y=key Write an equation in slopeintercept form
o Here is an example:
§ Write an equation of the line that passes through the point (3, 6) with a slope of 2.
· y = mx + b
· Substitute 2 for m, 3 for x, and 6 for y
· 6 = 2(3) + b
· 6 = 6 + b (Simplified)
· Solve for b
· 12 = b
· Rewrite in slope intercept form
· y = 2x + 12
· If you go back and plug in (3, 6), the left side should equal the right side!