The following equation models the height of a plant 𝑦 in millimeters 𝑥 weeks after Jenny purchases it from a local gardening store. 𝑦 equals 24𝑥 plus 60. When the equation is graphed in the 𝑥𝑦-plane, what does the slope of the graph represent in terms of the model?
Let’s think about what we know. If we consider this model in the 𝑥𝑦-plane, the 𝑥-axis will represent the weeks after Jenny has purchased the plant. And the 𝑦 represents the height of the plant in millimeters. We have this equation. 𝑦 equals 24𝑥 plus 60, which is in the form 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 represents the slope of the line. That means 𝑚 for us is 24. The slope is 24.
We know that the slope represents the changes in 𝑦 over the changes in 𝑥. In our model, the slope represents the change in height over the change in weeks. Since we have a slope of 24, we could write that as 24 over one. The height change is 24 per week. Since the height change is noted in millimeters, the slope is 24 millimeters per week, over one week. And so we can say that, in this model, the slope represents weekly plant growth of 24 millimeters.
If we wanted to see this on the graph, we could label our 𝑦-axis and our 𝑥-axis. If we plug in zero for 𝑥, we will see that 𝑦 equals 60. This means that the plant measured 60 millimeters when Jenny purchased it.
To find our second point, we could plug in one for 𝑥 to find out how tall the plant would be one week after Jenny purchased it. 24 times one is 24, plus 60 will equal 84. When 𝑥 equals one, 𝑦 equals 84. After that, we could try plugging in two. Two times 24 is 48, plus 60 equals 108. When 𝑥 equals two, two weeks after Jenny purchases her plant, the height will be 108 millimeters.
From there, we can connect these points. And then we see a change in 𝑦 of up 24 and a change in 𝑥 of right one, plus 24 over one. And we can look at the graph and see that this plant is growing at a rate of 24 millimeters every week.