How can linear functions be used as models

We already know the y-intercept of the line, so we can immediately write the equation:. Our model predicts the population will reach 15, in a little more than 23 years after , or somewhere around the year A company sells doughnuts.

In , the population was 28, By , the population was 36, It is useful for many real-world applications to draw a picture to gain a sense of how the variables representing the input and output may be used to answer a question. To draw the picture, first consider what the problem is asking for. Then, determine the input and the output.

The diagram should relate the variables. Often, geometrical shapes or figures are drawn. Distances are often traced out. If a right triangle is sketched, the Pythagorean Theorem relates the sides.

If a rectangle is sketched, labeling width and height is helpful. Anna and Emanuel start at the same intersection. Anna walks east at 4 miles per hour while Emanuel walks south at 3 miles per hour. They are communicating with a two-way radio that has a range of 2 miles. How long after they start walking will they fall out of radio contact?

In essence, we can partially answer this question by saying they will fall out of radio contact when they are 2 miles apart, which leads us to ask a new question:. In this problem, our changing quantities are time and position, but ultimately we need to know how long will it take for them to be 2 miles apart.

Using those values, we can write formulas for the distance each person has walked. For this problem, the distances from the starting point are important. Note that in defining the coordinate system, we specified both the starting point of the measurement and the direction of measure. This means that the distance between Anna and Emanuel is also a linear function. Because D is a linear function, we can now answer the question of when the distance between them will reach 2 miles.

Should I draw diagrams when given information based on a geometric shape? Sketch the figure and label the quantities and unknowns on the sketch. There is a straight road leading from the town of Westborough to Agritown 30 miles east and 10 miles north. Partway down this road, it junctions with a second road, perpendicular to the first, leading to the town of Eastborough.

If the town of Eastborough is located 20 miles directly east of the town of Westborough, how far is the road junction from Westborough? It might help here to draw a picture of the situation. It would then be helpful to introduce a coordinate system. While we could place the origin anywhere, placing it at Westborough seems convenient.

Using this point along with the origin, we can find the slope of the line from Westborough to Agritown:. We can now find the coordinates of the junction of the roads by finding the intersection of these lines. Setting them equal,. Using the distance formula, we can now find the distance from Westborough to the junction. The points do not appear to follow a trend. In other words, there does not appear to be a relationship between the age of the student and the score on the final exam.

The table below shows the number of cricket chirps in 15 seconds, for several different air temperatures, in degrees Fahrenheit. Plotting this data suggests that there may be a trend. We can see from the trend in the data that the number of chirps increases as the temperature increases. The trend appears to be roughly linear, though certainly not perfectly so. Finding the Line of Best Fit One way to approximate our linear function is to sketch the line that seems to best fit the data.

Then we can extend the line until we can verify the y -intercept. The resulting equation is represented in the graph below. This linear equation can then be used to approximate answers to various questions we might ask about the trend. While the data for most examples does not fall perfectly on the line, the equation is our best guess as to how the relationship will behave outside of the values for which we have data.

We use a process known as interpolation when we predict a value inside the domain and range of the data. The process of extrapolation is used when we predict a value outside the domain and range of the data.

The graph below compares the two processes for the cricket-chirp data addressed in the previous example. We can see that interpolation would occur if we used our model to predict temperature when the values for chirps are between Extrapolation would occur if we used our model to predict temperature when the values for chirps are less than There is a difference between making predictions inside the domain and range of values for which we have data and outside that domain and range.

Predicting a value outside of the domain and range has its limitations. When our model no longer applies after a certain point, it is sometimes called model breakdown. For example, predicting a cost function for a period of two years may involve examining the data where the input is the time in years and the output is the cost. Interpolation occurs within the domain and range of the provided data whereas extrapolation occurs outside.

Our model predicts the crickets would chirp 8. While this might be possible, we have no reason to believe our model is valid outside the domain and range.

In fact, generally crickets stop chirping altogether at or below 50 degrees. According to the data from the table in the cricket-chirp example, what temperature can we predict if we counted 20 chirps in 15 seconds? While eyeballing a line works reasonably well, there are statistical techniques for fitting a line to data that minimize the differences between the line and data values.

Least squares regression is also called linear regression. Notice also that using this equation would change our prediction for the temperature when hearing 30 chirps in 15 seconds from 66 degrees to:. Will there ever be a case where two different lines will serve as the best fit for the data? As we saw in the cricket-chirp example, some data exhibit strong linear trends, but other data, like the final exam scores plotted by age, are clearly nonlinear.

Most calculators and computer software can also provide us with the correlation coefficient , which is a measure of how closely the line fits the data. The correlation coefficient provides an easy way to get an idea of how close to a line the data falls. We should compute the correlation coefficient only for data that follows a linear pattern or to determine the degree to which a data set is linear.

If the data exhibits a nonlinear pattern, the correlation coefficient for a linear regression is meaningless. To get a sense of the relationship between the value of r and the graph of the data, the image below shows some large data sets with their correlation coefficients.

Remember, for all plots, the horizontal axis shows the input and the vertical axis shows the output. Plotted data and related correlation coefficients. Your calculator or software will provide you with the correlation coefficient when you use it to fit a linear regression. This value is very close to 1 which suggests a strong increasing linear relationship. Once we determine that a set of data is linear using the correlation coefficient, we can use the regression line to make predictions.

As we learned previously, a regression line is a line that is closest to the data in the scatter plot, which means that only one such line is a best fit for the data.

Gasoline consumption in the United States has been steadily increasing. Consumption data from to is shown in the table below. Use the model to predict the consumption in Is this an interpolation or an extrapolation? The correlation coefficient was calculated to be 0. The model predicts This is an extrapolation because there is not a datapoint whose x1 value is The scatter plot of the data, including the least squares regression line, is shown below.

Use your calculator or statistical software to find a linear regression for the following data, which represents the amount of time a scuba diver can spend underwater as a function of the depth of the water. Here are more data sets that you can plot in your calculator or statistical software. Fit a linear regression for them then interpre the correlation coefficient to determine whether there appears to be a linear relationship. Dimensions of the Lava Dome in Mt.

Divers who want or need to descend to depths greater than feet employ different techniques and equipment to help them safely navigate the depth. For example, different gas mixtures or rebreather equipment may be used. Gas mixtures such as oxygen, helium, and nitrogen can help to mitigate the narcotic effects of breathing gas at great depths. A scuba diver using rebreather with open circuit bailout cylinders returning from a foot m dive.

This point happens to be an intercept. The general formula for a line with these variables will have the form. The output will be Pay, and the input variable — the number of hours worked — will be hrs.

Thus, the equation of the line, according to the point-slope form is. This is a trick question part of the problem. From the text of the problem, the linear model only works for overtime, with a flat rate applying to less than 40 hours per week.

Comment: The function should be written as a piecewise defined function. It can be useful when writing reports to have variables that convey some meaning.

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All authors participated in writing and revising the manuscript and approved the final version. Correspondence to Mara Otten. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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