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Proportional relationships: graphs

Thuyết minh video

so I have three different relationships here between x and y and i want to think about which of these if any are proportional relationships and then i want to graph them to see if we can see anything visually that makes them obviously proportional and just as a reminder a proportional relationship is one where the ratio between the two variables and let's say we took the ratio between y and x you could also go the other way around the ratio between x and y but the ratio between y and x is always going to be some number some constant number or you could rewrite it another way if you were to multiply both sides of this equation times X you could say put in a proportional relationship Y is always going to be equal to some constant times X so with that out of the way let's look at these three relationships so this one over here let me do another column here another another column this is let me call this the Y over X column I'm just going to keep figuring out what this ratio is for each of these pairs so for this first pair when when X is 1 Y is 1/2 so this ratio is 1/2 over 1 well 1/2 over 1 is just the same thing as 1/2 when X is 4 y is 2 this ratio is going to be 2 over 4 which is the same thing as 1/2 when X is negative 2 and Y is negative 1 this ratio is negative 1 over negative 2 which is the same thing as 1/2 so for at least these 3 points that we sampled from this relationship it looks like the ratio between y and X is always 1/2 in this case K would be 1/2 we could write Y over X is always equal to 1/2 or at least for these 3 points that we sampled and we'll say well maybe it's always the case for this relationship between x and y or if you want to write it another way you could write that Y is equal to 1/2 X now let's graph this thing well when X is 1 Y is 1/2 when X is 4 y is 2 when X is negative 2 y is negative 1 I didn't put the marker for negative 1 would be but right about there and so if we say these three points are sampled on the entire relationship and the entire relationship is y is equal to 1/2 X well the the point the line that represents or the set of all points that would represent the the possible XY pairs it would be a line it would be a line that goes through the origin because look if x is 0 1/2 times 0 is going to be equal to Y and so let's think about some of the key characteristics one it is a line this is this is a line here it is a linear relationship and it also goes through the origin and it makes sense that it goes through an origin because if in a proportional relationship actually when you look over here is 0 over 0 that's indeterminate form and then that gets a little bit strange but when you look at this right over here well if X is 0 and you multiply it by some constant Y is going to need to be 0 as well so for any proportional relationship if you're including when x equals 0 then Y would need to be equal to 0 as well and also if you were to if you were to plot its graph it would be a line that goes through the origin it would be a line that goes through the origin and so this is a proportional relationship and it it's graph is represented by a line that goes through the origin now let's look at this one over here this one in blue so let's think about whether it is proportional and we could do the same test by calculating the ratio between y and X Y and X so it's going to be C for this first one is going to be 3 over 1 which is just 3 that's going to be 5 over 2 5 over 2 well 5 over 2 is not the same thing as 3 so already we know that this is not proportional not not proportional this is not a we don't have to look we don't even have to look at this Third Point right over here where if we took the ratio between y and X it's negative 1 over negative 1 which would just be 1 well let's see let's graph this just for fun and see what it looks like when X is 1 Y is 3 when X is 1 Y is 3 when X is 2 y is 5 X is 2 y is 5 and when X is negative 1 Y is negative 1 when X is negative 1 Y is negative I forgot to put the hash mark right there I was wearing it right around there and so if we said okay let's just give the benefit of doubt that maybe this is a these are three points from a line because it looks like I can actually connect them with a line then the line would look something like this the line would look something like the line would look something like this so notice this is linear this is a line right over here but it does not go through the origin so if you're just looking at a relationship visually linear is good but it needs to go through the origin as well for it to be proportional relationship and you see that right here this is a linear relationship or at least these three pairs could be sampled from a linear relationship but that but it does the graph does not go through the origin and we see here when we looked at the ratio that it wasn't indeed not proportional so this is not proportional now let's look at this one over here let's let's look let's look at what we have here so I'll look at the ratios Y over X so for this first pair 1 over 1 then we have 4 over 2 well we immediately see that we are not proportional and then 9 over 3 would be 3 so clearly this this is not a constant number here we don't always have the same value here so this is also not not proportional not proportional but let's graph it just for fun when X is 1 Y is 1 when X is 2 y is 4 this looks actually looks like the graph of this looks like the graph of y is equal to x squared when X is 3 y is 9 at least these three points are consistent with it so 1 2 3 4 5 6 7 8 9 so it's going to look something and so if this really is if these points are sampled from y equals x squared then when x is 0 Y would be 0 so this one actually would go through the origin but notice it's not a line it's not a linear relationship this right over here is the graph of y equals x squared so this one also is not proportional so once again these three points could be sampled from y equals 1/2 x these three points these three points could be sampled from let's see why is equal to let's see it looks like a line when this looks like it could be y is equal to 2x plus one so it's a linear relationship but it does not go through the origin so it's not proportional and this these three points looks like it could be look like they could be sampled from y equals x squared which goes through the origin when X is zero Y is zero but it's not a linear relationship but anyway you look at it if you look at it visually it has to be a line that goes through the origin or if you look at a table of values you just look at the ratios and the ratios always have to be the same value and that was only the case with this magenta one right over here