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let's get some practice interpreting graphs of proportional relationships so it says the proportional relationship between the distance driven and the amount of time driving is shown in the following graph so we have the distance driven on the vertical axis its measured in kilometers and then we have the time driving and its measured in hours along the horizontal axis and we can tell just visually that this indeed is a proportional relationship how do we know that well the point 0 0 is on this graph the graph goes through the origin if we have 0 time then we have 0 distance and we can also see that it's a line then it's a linear relationship if you have a linear relationship that goes through the origin you're dealing with a proportional relationship you could also see that by taking out some points here so let's see I'm just eyeballing it so I'm going to look at where does the graph kind of hit a very well defined points will actually point a right over here we see that when when our time is 5 hours our distance traveled is our distance driven is is what is this it's 400 kilometers 400 kilometers and then if we look at time C at this point right over here when our time is two-and-a-half when our time is two-and-a-half we see that our distance driven is 200 kilometers and notice the ratio between these variables at any one of these points is the same 400 divided by five is 80 and 200 divided by 2.5 is also going to be 80 or if you want to go the other way around you have to go from time to distance we're multiplying by or multiplying we're always multiplying by 80 in fact we could say the distance divided by time our proportionality constant is going to be 80 or if we wanted to include the unit's there it might be a little more obvious that we're dealing with the rate distance is in kilometers time is and hours 80 kilometers 80 kilometers per hour this is the rate at which we are driving we are going at this speed 80 kilometers per hour this is also the proportionality this is also the proportionality well anyway with all of that out of the way let's actually answer the questions which statements about the graph are true select all that apply the vertical coordinate of point a represents the distance driven in four hours the vertical coordinate the vertical coordinate so point a is at the coordinate 5 comma 400 now the vertical coordinate tells us how high to go up how far to move in the vertical direction that's going to be the second coordinate right over here so this is the vertical coordinate so this right over here tells us the distance we've driven in 4 hours so yes the vertical coordinate of point a represents the distance driven in 4 hours we've driven 400 kilometers so I like that one I'll check that one the distance driven in one hour is 80 kilometers the distance driven in one hour is 80 kilometers well when you just try to eyeball it off of this off of the graph here I see after one hour it looks like it's a little bit of more than 75 so yeah 80 seems like a reasonable one but we see it even more clearly when we look at the calculation that we did if you take a distance if you take a distance any one of these distance 400 kilometers 400 kilometers over five hours tells you that your rate is 80 is 80 kilometers 80 kilometers per hour and you know this is going to happen for any point on this line they tell us it's a proportional relationship and we can tell visually it's a proportional relationship it goes through the origin and it is a line so the distance traveled in one hour is going to be 80 kilometers it is going to be 80 kilometers because we're going 80 kilometers per hour so in one hour we're going to go 80 kilometers so I like that choice as well of course I won't pick none of the above because I found two choices that I I liked let's do another one of these this is fun a grocery store sells cashews the relationship between weight and cost of cashews is shown in the following graph and once again we see it as a proportional relationship it goes through the origin and it is a line which statements about the graph are true select all that apply the point zero zero shows the cost is zero dollars for zero kilograms of shoes yeah zero kilograms of cashews it's zero dollars yep that makes sense the point 2 comma 60 that's at this point right over here shows the cost is $2 for 60 kilograms of cashews the cost is $2 for 60 kilograms of cashews no this is showing us that the cost is $60 the cost is $60 for 2 kilograms for 2 kilograms of cashews this axis right over here this is weight actually if you're measuring kilograms are really talking about mass but anyway I don't want to get too particular here but this you have a mass here of 2 kilograms so the the I guess you say the horizontal coordinate tells you the mass the vertical coordinate tells you the cost and that is 60 $60 so it's $60 for 2 kilograms not $2 for 60 credit that would be a deal if you could get if you could get you know this is kind of what a rediem size grown man would weigh 60 kilograms some $2 for that much that much cashews that's that would be an incredible deal so you definitely want to rule that one out and I'm not going to pick none of the above because I already figured out that I liked I liked the first choice up there let's let's just do one more just for good measure an employee earns an hourly wage shown in the graph below find the hourly rate wage so we see here in the horizontal axis we have hours worked and in the vertical axis earnings we see it's a proportional relationship that the that the and that makes sense the ratio between earnings and hours should always be constant if we're earning money at a constant rate we see it visually we go through the origin and we're dealing with a line so there's a bunch of ways that we could look at it we could say we could say hours so let's see hours hours and then dollars after 1 hour we've made $40 and this effect visually that we know it's a proportional relationship says hey we're making $40 per hour 40 if we take dollars divided by hours I'll write it out the unit's $40 $40 divided by 1 hour this right over here is $40 per our you could view this as our proportionality constant or you could use the rate in which we're earning money forty dollars per hour and you see that two hours eighty dollars two hours eighty dollars eighty divided by two is 43 hours 120 dollars three hours 120 dollars 120 divided by three is 40 our proportionality constant is 40 that your earnings your earnings earnings divided by hours hours is always going to be equal to 40 or you could say that your earnings your earnings is equal to if you multiply both sides by hours is going to be equal to 40 times your hours your earnings in dollars is going to be equal to 40 times your hours