6.10 Fitting functions to exponential data
As we saw in our lesson on quadratic regressions, sometimes we need to consider fitting something other than a linear model to data. In Algebra 1, we saw that we could manipulate data to fit a linear model to it by adjusting to make it more linear.
Since we have excellent technology at our disposal, we can leave the data as is and instead fit a different regression model to our data if we need to.
Using technology to fit regression models
Let's consider the following data set from the above-mentioned quadratic regressions section.
| $x$x | $4$4 | $4.8$4.8 | $5.1$5.1 | $6$6 | $7.1$7.1 | $8.2$8.2 | $9.4$9.4 |
|---|---|---|---|---|---|---|---|
| $y$y | $19.4$19.4 | $20.4$20.4 | $20.2$20.2 | $19.1$19.1 | $18$18 | $14.9$14.9 | $10$10 |
As we can see from the scatter graph, the data looks parabolic.
Instead of transforming the data, let's have our calculator fit a quadratic regression model to the data.
As you can see, when I go to choose a model for regression, there are many to choose from. Your knowledge of functions will help you make the best choice. Sometimes the data will fit a linear or quadratic function, but sometimes the data will look like an exponential or linear function.
And here we have the equation of the quadratic function fitted to the data.
We can see the value of the coefficient of determination, $r^2$r2, is very strong, and we have the quadratic function $y=-0.52x^2+5.27x+6.86$y=−0.52x2+5.27x+6.86.
Practice questions
Question 1
Question 2
When CTech first released a digital application (an ‘app’) onto the market, the number of sales increased slowly at first, but then the number of sales started to increase very rapidly.
Which scatter plot shows the trend in sales over time from when the app was first released?
Loading Graph...ALoading Graph...BLoading Graph...CThe function $y=1000\left(50^{\frac{t}{10}}-1\right)$y=1000(50t10−1) is used to approximate the number of sales after $t$t months, where $y$y represents the number of sales.
Complete the table of values for $y=1000\left(50^{\frac{t}{10}}-1\right)$y=1000(50t10−1).
$t$t $0$0 $10$10 $y$y $\editable{}$ $\editable{}$ $20$20 months after CTech released their app, a rival company, BTech, released a similar app with improved features. In the month that followed, CTech’s sales dropped by $60000$60000 from their previous month's sales. According to the model $y=1000\left(50^{\frac{1}{10}t}-1\right)$y=1000(50110t−1), what were CTech’s sales one month after BTech released their new app?
Make predictions from a graph
Given a set of data relating two variables $x$x and $y$y, we can use a model to best estimate how the dependent variable $y$y changes in response to the independent variable $x$x. The model allows us to go one step further and make predictions about other possible ordered pairs that fit this relationship.
Exploration
Say we gathered several measurements on the population $P$P of fish in a small lake over $t$t years. We can then plot the data on the $xy$xy-plane as shown below.
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| Population of fish in a lake measured at several instances. |
We can fit a model through the observed data to make predictions about the population at certain times. One plausible model might be the following exponential function.
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| Exponential curve modeling population of fish in a pond over time. |
To make a prediction on the population, say two years later, we first identify the point on the curve when $t=2$t=2. Then we find the corresponding value of $P$P. As you can see below, the model predicts that two years after the earthquake, the number of fish was $250$250.
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| A predicted population of $250$250 fish when $t=2$t=2. |
If we predict the population six years after the earthquake, we find that the population is roughly $16$16. A prediction outside the observed data set such as this one is called an extrapolation.
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| A predicted population of $16$16 fish when $t=6$t=6. |
How reliable are these predictions? Well, any model that fits the observed data will make reliable predictions from interpolations since the model roughly passes through the center of the data points. We can say that the model follows the trend of the observed data.
However extrapolations from outside the given data points are generally unreliable since we make assumptions about how the relationship continues outside of collected data. Sometimes extrapolation can be made more reliable if we have additional information about the relationship.
Practice questions
Question 3
To investigate the environmental effect on bacterial growth, two colonies of the same bacteria were placed one in a constantly sunlit environment, the other in a dark environment. The graph shows the population of each colony after a certain number of days.
How many more bacteria were initially present in the sunlit environment?
By what percentage rate did the bacteria left in sunlight increase each day?
By what percentage rate did the bacteria left in darkness increase each day?
Initially there were more bacteria in the sunlit environment. Why then do the two curves on the graph meet?
The population of bacteria in the dark environment is decreasing at a slower rate than the population of bacteria in the sunlit environment.
AThe population of bacteria in the sunlit environment is decreasing at a slower rate than the population of bacteria in the dark environment.
BThe population of bacteria in the sunlit environment is increasing at a faster rate than the population of bacteria in the dark environment.
CThe population of bacteria in the dark environment is increasing at a faster rate than the population of bacteria in the sunlit environment.
D
Question 4
Switzerland’s population in the next $10$10 years is expected to grow approximately according to the model $P=8\left(1+r\right)^t$P=8(1+r)t, where $P$P represents the population (in millions) $t$t years from now.
The world population in the next $10$10 years is expected to grow approximately according to the model $Q=7130\left(1.0133\right)^t$Q=7130(1.0133)t, where $Q$Q represents the world population (in millions) $t$t years from now.
According to the model, what is the current population in Switzerland?
According to the model, what is the current world population?
Switzerland’s population is expected to increase at a slower rate than the world population. Switzerland’s population increase each year could be:
$1.79%$1.79%
A$0.87%$0.87%
B