Graph labels and scales | Modeling | Algebra II | Khan Academy

We're told that Chloe takes a slice of pizza out of the freezer and leaves it on the counter to defrost. She models the relationship between the temperature ( p ) of the pizza, this seems like it's going to be interesting. The temperature ( p ) of the pizza in degrees Celsius and time ( t ) since she took it out of the freezer in minutes as ( p = 20 - 25 \cdot 0.8^t ). So that's how she's modeling her temperature of the pizza ( p ) as a function of time.

She wants to graph the relationship over the first 25 minutes. So what we're going to do here is not so much focus on the graph itself, although we will look at that. I'm actually just going to use a graphing calculator in order to have access to the graph, but I want to look at the graph in the context of what we are trying to model. We'll carefully think about what should be the labels for the axes and what parts of the graph are interesting.

So this right over here is this function graphed on Desmos. You can see I typed it in right over here: ( p = 20 - 25 \cdot 0.8^t ), exactly what we had down here. Now, remember this is modeling the temperature of our pizza as a function of time. So to help us remember that, let's put in some labels for our axes.

In the graph settings, if I go down here, our ( x )-axis now, our ( x )-axis is really the ( t )-axis. That's our independent variable, and what is it measuring? Well, it says it right over here: it's measuring time ( t ) in minutes. So we could write it like this: ( t ) which is measured in minutes.

Then what about our ( y )-axis? Well, this is really our ( p )-axis, and that's measuring degrees Celsius. So that's our ( p )-axis, and it's measuring degrees Celsius ( (^{\circ}C) ).

All right, so let's just look at what our graph looks like so far. So there we have it; we've put in our axes and we have already typed this part in, so I can make it so I can focus on the graph itself. Now, are we done? Is this all we need to really think about?

Well, the next part to think about is the domain and what part of the ( y )-axis, what part of the range are we really interested in? Well, the first thing to realize is we're modeling something as a function of time. So we really shouldn't be having negative time here, and we want to think about the relationship over the first 25 minutes.

So let's go back here, and when we look at the range of ( x ) values that we care about, we could think about the part of the domain that we care about. We want to restrict ( x ) to being greater than or equal to zero, and obviously, in this situation, ( x ) is really ( t ). Then we can also think about it as less than 25.

We don't have to restrict the upper bound, but these are the first 25 minutes that she cares about. So let's do it like that. Now, let's look at our graph, and the important things to appreciate are that we have the axes, we can see them.

So at time ( t = 0 ), we see that we actually have a negative temperature in degrees Celsius, and that makes sense; it came out of the freezer, so it's below freezing. Then we see that the pizza is warming up as it gets closer and closer to room temperature, which over here looks like it's pretty close to 20 degrees Celsius.

Now it looks like we have been able to graph what Chloe is trying to look at. It looks like we have modeled it well, we have labeled it accordingly, and we have set the ranges of ( x ) values and the ranges of ( y ) values that we'd want to look at. The ( y ) values, we just want to make sure that over the range of ( x ) values, it's really a subset of the domain—not to confuse the term "range" too much—the subset of the domain of the ( x ) values that we care about that we can see the corresponding ( y ) values.

And we very clearly can see them, and we're essentially done. We've thought about how to best look at the graph of this model.