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### Course: Algebra 2 > Unit 12

Lesson 4: Modeling with two variables# Graph labels and scales

When graphing a real-world relationship, we need to pick labels and axis scales that are appropriate for the purpose of our model. Created by Sal Khan.

## Want to join the conversation?

- to me i see no way you could use it like this in everyday life except, possibly your job(9 votes)
- This will be very important in certain jobs, but you could use it in everyday life too. If you want to compare certain teams and their wins, you might need to use this. This is useful.(7 votes)

- I have another problem that I can't figure out. It goes like this:

Ashley is doing some math exercises on a website called Khan Academy. In Khan Academy, you have to get at least 70% of the problems in an exercise right in order to gain proficiency.

So far, Ashley has answered correctly 3 out of 7 times. Suppose she answers all of the following q questions correctly and gains proficiency in the exercise.

Write an inequality in terms of q that models the situation.

None of it makes sense to me. How are you supposed to do this problem?(4 votes)- Let q = questions answered correctly
**Key Terms**:

- At least get 70%

- Ashley already answered 3

- Assume Ashley solves all of the questions correctly after answering 3

"*at least 70% of the problems in an exercise right*"

Translated: .7 <= [expression]

"*So far, Ashley has answered correctly 3 out of 7 times.*"**and**"*Suppose she answers all of the following q questions correctly and gains proficiency in the exercise.*"

Ashley already got 3/7 correct ! Excellent, and now she solved**q**questions correctly.

Imagine this:

[correct questions]/[total questions]

Correct questions would be 3+**q**

Total questions would be 7+**q**, because we don't know how many questions she answered ! We just know that she answered**q**questions correctly**after**getting 3 questions correct.

Our inequality now is:

.7 <= (3+q)/(7+q)

hopefully that helps !(4 votes)

- How did Chloe model her graph as P=20-25*(0.8)^t? How did she derive (0.8)^t? And why didn't she derive P= 0-25 degrees instead of 20-25? I'm just curious how to use these parameters sometime for my own experiment.

Thanks in advance.(2 votes)- Unfortunately, this is just a model non-representative with real life for the sake of explaining exponential models. However though in exponential equations:
**a(b)^x+c**

a = Initial Value

b = Constant trend growth/decay

x = Time (determined by how you define it

c = Horizontal Asymptote (as you increase x, the output will level out to this value(7 votes)

- Why does the graph plateau at 20. How would I know by looking at a exponential equation when it would plateau?(2 votes)
- This is because the graph is exponential, and the pizza is slowly becoming warmer to the room temperature of 20 degrees Celsius.(1 vote)

## Video transcript

- [Instructor] 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 is equal to 20 minus 25 times 0.8 to the T power. 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 wanna look at the
graph in the context of what we are trying to model and carefully think about
what should be the labels for the axes, what parts of the graph are interesting? So this is right over here is this function
graphed on Desmos. You can see I typed it in right over here. P is equal to 20 minus 25 times 0.8 to the T power. 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. So to graph, settings. If I go down here, our x-axis. Now our x-axis is really the T-axis. That's our independent variable over here. 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. And 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. 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 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. So 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. And so we really shouldn't
be having negative time here. And we wanna 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, and really that we could think
about the part of the domain that we care about, we wanna restrict to x being greater than or equal to zero. And obviously in this situation, x is really T and 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. And now let's look at our graph. And the important things to appreciate is that we have the axes. We can see them. And so at time T is equal to zero, 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. And 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. And so 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 wanna look at. The y values we just wanna
make sure that over the range of x values, and 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.