Mathew rides his bike at a steady rate of 15 miles per hour. The function 𝑦 equals 15𝑥 shows how time 𝑥 relates to the distance 𝑦. Which graph represents this function?
Well, first of all, we can say that the time is 𝑥 and the distance is 𝑦. So therefore, straight away, we can rule out graphs C and E. And that’s because they actually have their axes the other way, because we have our time at 𝑦-axis and we have our distance as our 𝑥-axis.
So now we actually have three graphs left to choose from. And to enable us to do that, what we’re gonna have a look at is the fact that this is actually Matthew riding his bike at a steady rate. And therefore, as he’s actually riding his bike at a steady rate, we know this is gonna be a linear graph. Well, this doesn’t rule any graphs out. But it does give us something important, because we know that the equation of a straight line is 𝑦 equals 𝑚𝑥 plus 𝑐, given that 𝑚 is the slope and 𝑐 is the 𝑦 intercept.
Well, if we look at our function, which is 𝑦 equals 15𝑥, first of all, we can see that there’s no 𝑐. So therefore, the 𝑦-intercept must be zero. Well, for the three graphs that are left, the 𝑦-intercept is actually zero on each of these, so doesn’t rule any out. But next, we’re gonna look at the slope. And the slope of our function is 15. So therefore, we need to find the graph that has a slope of 15.
So the slope is equal to change in 𝑦 divided by change in 𝑥. Okay, so great! We now know what the slope is. Let’s try and find out whether any of our graphs have a slope of 15. Now because each of our graphs actually goes through the origin, it’s gonna be very easy for us to actually find out the slope of our graphs. As if they actually have a slope of 15, then they’re gonna go through the point one, 15. But if we look at graph A, we can see if we go one on the 𝑥-axis, then we get 15 on the 𝑦-axis. So this could be our graph. And it probably will be.
But we’re gonna have a look at B and D just to make sure. But if we look at graph B and we go up at one in the 𝑥-axis, we actually get below 15 on the 𝑦-axis. So therefore, our slope will not be 15. And we can double-check this by using an 𝑥 value of two because it should give us a 𝑦 value of 30. And it doesn’t, actually gives us around 15. So therefore, this definitely does not have a slope of 15 and therefore cannot be our graph.
So finally, we’re gonna look at graph D. And if we look at graph D, this is much to state because actually our 𝑥 value of one gives us a value much higher than 15. So therefore, the slope would be greater than 15. And therefore, we can say this isn’t our graph. So therefore, we can say that graph A represents the function 𝑦 equals 15𝑥.