Find the rate of change for the following graph.
Our graph is talking about temperature change. On the 𝑦-axis, we’ve the temperature in degrees Fahrenheit. And on the 𝑥-axis, we’ve the time in hours.
So looking at our graph, we can see that after one hour, our temperature has decreased. And after two hours, it’s decreased even more. And after more hours go by, the temperature keeps decreasing. So we want to find the rate that it’s decreasing.
And this rate of change is actually the slope of our graph, the gradient, called 𝑚. And it’s the change in 𝑦-values divided by the change in 𝑥-values, sometimes called the rise over the run. And the way that we can determine 𝑚 is to look at two points on our graph. So let’s pick two points that the graph for sure goes through, that are exact points.
Well, we don’t have to guess where they are. We could use this point here at zero, 60 and this point here at six, 30. So zero and six are 𝑥-values. And 60 and 30 are our 𝑦-values. So we go from zero, 60 to six, 30. So what was the change in the 𝑦s? We went from 60 to 30. That would be a decrease of 30. And then on the denominator, the change in the 𝑥-values, we go from zero to six, which would be positive six. And negative 30 over six would be negative five over one.
We actually could’ve used other values too. Maybe instead, we used these points two, 50 and four, 40. The change in the 𝑦-values would be 50 to 40. So it decreased 10. And then the change in the 𝑥-values was from two to four. It increased two. And negative 10 over two reduces to negative five over one.
So we could write our rate of change as negative five over one. Or using words, we could say that the temperature decreases five degrees every one hour.