Worked example: slope from graph

find the slope of the line in the graph and just as a bit of a review slope is just telling us how steep a line is and the best way to view it slope is equal to change in Y change in Y over change in x over change in X and for a constant this will all or sorry for a line this will always be constant and sometimes you might see it written like this you might see this triangle that's a capital Delta that means change in change in Y over change in X that's just a fancy way of saying change in Y over change in X so let's see what this change in Y is for any change in X so let's start at some point that seems pretty reasonable to read from this table right here from this graph so let's see we're starting here let me do it in a more vibrant color so let's say we start at that point right there and we want to go to another point that's pretty straightforward to read so we can move to that point right there we could literally pick any two points on this line I'm just picking ones that are at nice integer coordinates so it's easy to read so what is the change in Y and what is the change in X so first let's look at the change in X so if we go from there to there if we go from there to there what is the change in X my change in X is equal to what well I could just count it out I'll go I went one step two steps three steps my change in X is three and you can even see it from the x-values if I go from negative 3 to zero I went up by three so my change in X is three so let me write this change in X Delta X is equal to three and what's my change in Y well my change in Y I'm going from negative three up to negative one or you could just say you could say one two so my change in Y my change in Y is equal to positive two let me write that down change in Y is equal to two so what is my change in Y for a change in X well when X when my change in X was 3 my change in Y is 2 so this is my slope and one thing I want to do I want to show you that I could have really picked any two points here let's say I didn't pick let me clear this out let me clear let's say I didn't pick those two points let me pick some other points and I'll even go in a different direction I want to show you that you're gonna get the same answer let's say I see this is my starting point and I want to go all the way over there so what is my let's think about the change in Y first so the change in Y I am going down I am going down by how many how many units 1 2 3 4 units so my change in Y in this example is negative 4 I went from 1 to negative 3 that's negative 4 that's my change in Y change in Y is equal to negative 4 now what is my change in X well I'm going from this point or from this x value all the way let me do that in a different color all the way all the way back like this so when to the left it's going to be a negative change in X and I went 1 2 3 4 5 6 units back so my change in X is equal to negative 6 change in X is equal to negative 6 and you could even see how when it started at an X is equal to 3 and I went all the way to X is equal to a negative 3 that's a change of negative 6 I went I went 6 to the left or a change of negative 6 so what is my change in Y over change in X my change in Y over change in X is equal to negative 4 over negative 6 the negatives cancel out and what's 4 over 6 well that's just 2 over 3 so it's the same value you just have to be consistent if this is my start point I went down 4 and then I went back 6 negative 4 over negative 6 if I view this is my starting point I could say that I went that I went up four that I went up 4 so it would be a change in Y would be 4 and then my change in X my change in X would be 6 and either way once again change in Y over change in X is gonna be four over six two-thirds so no matter which point you choose as long as you kind of think about it in a consistent way you're gonna get the same value for slope