Which of the given graphs represents the direct variation between 𝑥 and 𝑦?
And then we have four different graphs to choose from. So, in order to be able to identify the relevant graph, we need to remind ourselves what we mean by direct variation or direct proportion. This symbol that looks a little bit like the Greek letter 𝛼 but isn’t represents direct proportion. This statement means 𝑦 is directly proportional to 𝑥. If 𝑦 is directly proportional to 𝑥, then this means that their ratio is constant. 𝑦 divided by 𝑥 is always equal to some value 𝑘. We can alternatively represent this as 𝑦 is equal to 𝑘 times 𝑥. 𝑘 is called the constant of proportionality, or the constant of variation.
What this also in turn means is that one of the criteria for two variables to be in direct proportion to one another is that when one is equal to zero, the other is also equal to zero. And this makes a lot of sense. If we compare the equation 𝑦 equals 𝑘𝑥 to the equation for a straight line given in slope–intercept form, the equation in this form is 𝑦 equals 𝑚𝑥 plus 𝑐. 𝑐, of course, is the value of the 𝑦-intercept. But if when 𝑥 is equal to zero, 𝑦 is equal to zero, the 𝑦-intercept, in fact, is zero. So, we get that equation in the form 𝑦 equals 𝑘𝑥.
But let’s think about what the value of 𝑚 means in the equation 𝑦 equals 𝑚𝑥 plus 𝑐. This is the slope. It tells us how steep the line itself is. So, that’s what the value of 𝑘 means when we think about this graphically. In fact, we don’t really even need this information to answer the question. We need to find a graph that passes through the origin, zero, zero.
We notice graph (a) does not pass through the origin, graph (b) does, graph (c) does not, nor does graph (d). And so, graph (b) represents direct variation between 𝑥 and 𝑦. Specifically, since the graph slopes downwards, it represents a situation where the constant of variation is negative; 𝑘 is less than zero. The answer is (b).