6.07 Linear and exponential models (M)

A real estate agent earns 3\% of the value of every house sold.

Creating a model will help determine if this situation is linear or exponential.

Let x represent the value of the house sold. Then, the amount earned by the real estate agent can be modeled with f(x)=0.03x.

This situation is better modeled with a linear function because it has a constant rate of change.

We can calculate the amount the agent earns by finding 3\% of the house value. If we put a few different house values in a table, we get

which shows us that the agent earns \$3000 for every \$500\,000 of house sold. This is a linear rate of earnings.

We can also graph the earnings dependent on house value to see that they form a line:

The average median house price of a home, y, sold in the U.S. from 2019 to the beginning of 2022 is shown in the graph where x represents the number of years since 2019.

We're given this context on a graph so we can consider what it would look like to draw a line versus a curve through the points.

The graph appears to be curved similarly to exponential growth, so we will choose exponential growth for our model.

The small amount of change in 2019 is similar to the horizontal asymptote of an exponential function with a vertical translation of about 375\,000. If we draw an exponential curve through the data, it might look like this:

To verify that a linear function is not a better fit, we can see that a line would not be near a majority of the points:

Create a linear model to represent the average annual tuition cost for each type of university.

A linear model will have a constant slope. We can find the average rate of change from 1980 to 2020 by using the average rate of change formula f(x)=\dfrac{f(b)-f(a)}{b-a}.

The average rate of change from 1980 to 2020 for public university average annual tuition was \dfrac{9403-1856}{2020-1980}=\$188.68 If we let x represent the years since 1980, we can use the slope-intercept form to get: y=188.68x+1856

We can see from the graph that this model overestimates the tuition from 1990 to 2010.

The average rate of change from 1980 to 2020 for private university average annual tuition was: \dfrac{34\,059-10\,227}{2020-1980}=\$595.8 If we let x represent the years since 1980, we can use the slope-intercept form to get: y=595.8x+10\,227

We can see from the graph that this model is a pretty good fit except for in 2000.

A better model for public university tuition and fees might be the line of best fit: f(x)=181.58+1074.2, which has a similar slope, but a lower y-intercept

Create an exponential model to represent the average annual tuition cost for each type of university.

An exponential model in the form f(x)=ab^x has an initial value and a growth factor. We can find the growth factor by finding the ratio of tuition costs for two different points.

The growth factor from 1980 to 2020 for public university average annual tuition was \dfrac{9403}{1856}=5.066 over a span of 40 years. We can use properties of exponents to find the annual growth factor for our model: (5.066)^\frac{1}{40}=1.041. This tells us that the tuition increased an average of 4.1\% each year from 1980 to 2020. If we let x represent the years since 1980, we can create an exponential model:y=1856(1.041)^x

We can see from the graph that this model closely follows the data.

The growth factor from 1980 to 2020 for private university average annual tuition was: \dfrac{34\,059}{10\,227}=3.330 We can use properties of exponents to find the annual growth factor for our model: (3.330)^\frac{1}{40}=1.031. This tells us that the tuition increased an average of 3.1\% each year from 1980 to 2020. If we let x represent the years since 1980, we can create an exponential model:y=10\,227(1.031)^x

We can see from the graph that this model underestimates tuition in 1990 and 2000.

Determine if a linear model or an exponential model would be better to predict tuition for the next decade. Explain the differences between the models and what they tell us about the context.

We have linear models in part (a) and exponential models in part (b). We should choose our model based on how well they fit the current data and if we think the trend will continue into the next decade.

The exponential model looked to be a better fit for public university tuition, and the linear model looked to be a better fit for private university tuition.

Using y=1856(1.041)^x, to predict public school tuition in the next ten years, we find that the average annual public university tuition would be about y=1856(1.041)^{50}=\$13\,839.31.

Using y=595.8x+10\,227, to predict private school tuition in the next ten years, we find that the average annual private university tuition would be 595.8(50)+10\,227=\$40\,017.

The change in exponential models increases over time, but linear models stay consistent. Since the private university tuition had a more linear growth, we might expect their tuition to continue that trend. The public university tuition experienced larger and larger increases each decade which is more consistent with exponential growth. If these trends continued, we would see public university tuition catch up to private university tuition.

It's more likely that private university tuition will increase exponentially as the cost of goods also increases by a percent each year. However, if the trends continue the way we've seen them, here's what tuition would like from 1980 to 2080: