Introduction Data surrounds us in the real world. Every day, people are presented with numbers and are expected to make predictions about future events

IntroductionData surrounds us in the real world. Every day, people are presented with numbers and are expected to make predictions about future events based upon that given data. A regression equation is an equation that best represents a set of data, and it can be used to predict missing data or future data. Different types of equations are suited to different types of data. Regression is the mathematical process for determining an equation from a set of given data. Regression is used to make predictions for values of an independent variable. Some data is best represented by linear or exponential equations, as you have seen previously.

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5.9.1: Solving Problems Given Functions Fitted to Data

Introduction, continuedQuadratic regression is the process of finding the equation of a parabola that fits a given set of data. In this lesson, you will work with data sets that are best represented by quadratic equations, and you will learn how to write a quadratic regression equation, a regression equation that fits a parabola to data.

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Key Concepts• A linear equation describes a situation where there is

a near-constant rate of change. • An exponential equation describes a situation where

the data changes by a constant multiple. • A quadratic equation describes data that increases

then decreases, or vice versa. • If you are given a set of data and you are not sure

whether the data is best modeled by a linear regression or a quadratic regression, you can look at the first and second differences.

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Key Concepts, continued• In a linear model, the y-value changes by a constant

when the x-value increases by 1. The change in y when x increases by 1 is called a first difference. If your first differences are all about the same, then a linear model is appropriate.

• In a quadratic model, the first differences are not the same, but the change in the first differences is constant. The change in successive first differences is called a second difference.

• A quadratic regression equation fits a parabola to the data.

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Key Concepts, continued• The regression equation closely models the data but

is not necessarily an exact fit. Actual data values and regression values might differ.

• Regression equations can be used to make predictions about the dependent variable for given values of the independent variable.

• Interpolation is when a regression equation is used to make predictions about a dependent variable that is within the range of the given data.

• Think of interpolation as finding “missing” data points within the given data.

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Key Concepts, continued• To interpolate, substitute the x-value into the given

regression equation and solve for the y-value. • Extrapolation is when a regression equation is used

to make predictions about a dependent variable that is outside the range of the given data.

• Think of extrapolation as predicting data values based on the model outside of the given data.

• To extrapolate, substitute the x-value into the regression equation and solve for the y-value.

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Key Concepts, continued• The farther away you move from the given data to

make predictions, the less accurate your predictions become.

• Be careful when extrapolating data and always make sure the predictions are reasonable.

• To write a regression model for a set of data without a calculator, first plot the given points. If the data has a basic parabolic shape (the values rise and then fall, or vice versa), you can write a quadratic equation to model the data.

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Key Concepts, continued• To write a quadratic regression equation, determine

where the vertex would be for a parabola that models your data. Then, use either the two x-intercepts or the y-intercept to write the equation.

• Graph your regression equation to make sure that it approximates the data.

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Common Errors/Misconceptions• confusing which type of function best describes the

data • forgetting to consider whether extrapolations are

reasonable • confusing the independent and dependent variables • choosing the incorrect regression model for the given

data

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Guided Practice

Example 1The following data table shows a car’s speed in miles per hour and the car’s fuel efficiency in miles per gallon for each speed.

A quadratic regression equation that models this data is given by m(x) = –0.0146x2 + 1.1802x + 9.1356, where x is speed in mph and m(x) is fuel efficiency in mpg.

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Speed (mph) 18.6 24.9 31.1 37.3 43.5 49.7 55.9 62.1Fuel efficiency (mpg) 26.1 29.4 31.4 33.1 33.1 31.4 29.4 26.1

Guided Practice: Example 1, continuedA scatter plot of the data with the graph of this model is shown to the right.

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Guided Practice: Example 1, continuedUse the given regression model to find the car’s fuel efficiency in miles per gallon when this car is traveling 31.1 mph. Compare your answer to the data in the table. Do these values match? Then use the graph to estimate the speed(s) that will result in fuel efficiencies of about 25 mpg and 40 mpg. Use the model to check your estimates.

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Guided Practice: Example 1, continued

1. Use the regression model to find the fuel efficiency in miles per gallon when this car is traveling 31.1 mph. Substitute 31.1 for x in the given equation.

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According to the regression model, a car traveling 31.1 mph would get approximately 31.7186 miles per gallon.

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Quadratic regression equation

Substitute 31.1 for x.

Solve.


2. Compare your answer to the data in the table. The data in the table indicates that a car traveling 31.1 mph gets 31.4 mpg.

These values do not match exactly, but they are very close. A regression model does not necessarily pass through every data point, so individual values may be slightly different when the model and the actual data are compared. However, it is still a good model for representing the data.

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3. Use the graph to estimate the speed(s) that will result in fuel efficiency of about 25 mpg. Use the model to check your estimates. Graph a horizontal line at y = 25.

The horizontal line indicates where the mpg is equal to 25. This line crosses the parabola twice. Those x-values appear to be about 17 mph and 64 mph.

The graph is shown on the next slide.

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4. Use the model equation to check your estimate. Since 25 is the y-value, set the given equation equal to 25.

–0.0146x2 + 1.1802x + 9.1356 = 25

Next, subtract 25 from both sides to set the equation equal to 0.

–0.0146x2 + 1.1802x – 15.8644 = 0

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Guided Practice: Example 1, continuedNow that the quadratic equation is set up in the form ax2 + bx + c = 0, determine the values of a, b, and c.

a = –0.0146, b = 1.1802, and c = –15.8644

Substitute these values into the quadratic formula,

, and solve the resulting equation

for x.

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Simplify, then solve.


Substitute values for a, b, and c:

Guided Practice: Example 1, continuedThe quadratic formula yields x-values of approximately 17.0 mph and 63.8 mph. These values are a good match for the values estimated using the graph.

At speeds of about 17.0 mph and 63.8 mph, we can expect to attain a fuel efficiency of about 25 mpg.

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5. Use the graph to estimate the speed(s) that will result in a fuel efficiency of about 40 mpg. Graph a horizontal line at y = 40 (see the next slide).

The horizontal line on the graph at y = 40 indicates at which speeds the car gets 40 miles per gallon. The maximum of the parabola is below this line. Therefore, this vehicle cannot achieve a fuel efficiency of 40 mpg at any speed.

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6. Use the model equation to check your result. Since 40 is the y-value, set the given equation equal to 40.

–0.0146x2 + 1.1802x + 9.1356 = 40

Subtract 40 from both sides to set the equation equal to 0.

–0.0146x2 + 1.1802x – 30.8644 = 0

Find the discriminant to see if a solution exists.

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Guided Practice: Example 1, continuedThe formula for the discriminant is d = b2 – 4ac.

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a = –0.0146, b = 1.1802, and c = –30.8644

d = (1.1802)2 – 4(–0.0146)(–30.8644)

Substitute values for a, b, and c.

d = 1.39287204 – 1.80248096 Simplify, then solve.

d ≈ –0.41


The discriminant is negative. This means that, according to the model, it is not possible for this particular car to get 40 mpg at any speed.

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Guided Practice

Example 2Use the regression model and graph from Example 1 to find the x- and y-intercepts of the graph. Interpret their meanings. Then, use the equation to predict the car’s fuel efficiency at the speeds of 20 mph, 65 mph, 75 mph, and 90 mph. Determine whether each of these predictions is an interpolation or an extrapolation, and whether any of the predictions seem unreasonable within the context of the problem.

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Guided Practice: Example 2, continuedThe following data table from Example 1 shows a car’s speed in miles per hour and the car’s fuel efficiency in miles per gallon for each speed.

A quadratic regression equation that models this data is given by m(x) = –0.0146x2 + 1.1802x + 9.1356, where x is speed in mph and m(x) is fuel efficiency in mpg. A scatter plot of the data with the graph of this model is shown on the next slide. 29


Speed (mph) 18.6 24.9 31.1 37.3 43.5 49.7 55.9 62.1Fuel efficiency (mpg) 26.1 29.4 31.4 33.1 33.1 31.4 29.4 26.1


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1. Find the x-intercepts of the graph. The equation is –0.0146x2 + 1.1802x + 9.1356.

Solve this equation for x using the quadratic formula,

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The quadratic formula yields x-values of approximately –7.11 and 87.95 mph.

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Simplify, then solve.

Substitute values for a, b, and c.


2. Interpret the meaning of the x-intercepts.

The x-intercepts are x ≈ –7.11 and x ≈ 87.95. According to the model, these are the speeds that have a fuel efficiency of 0 mpg. A car cannot achieve negative miles per hour; therefore, –7.11 does not make sense in the context of the problem. At 87.95 mph, a car is still moving, so it would still be using gas; therefore, a fuel efficiency of 0 mpg does not make sense within the context of the problem. The model appears to be valid only for data points near the values given in the data table. 33



3. Find the y-intercept. Substitute 0 for x to find the y-intercept.

The y-intercept is 9.1356.34


Quadratic regression equation

Substitute 0 for x.

Solve.


4. Interpret the meaning of the y-intercept.

The y-intercept is 9.1356. This means that the car gets about 9.1 mpg when it is not moving. The only way for this to make sense is if the car is idling.

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5. Use the equation to predict the fuel efficiency at the speeds of 20 mph, 65 mph, 75 mph, and 90 mph. Determine whether each of these predictions is an interpolation or an extrapolation, and if any of the predictions seem unreasonable within the context of the problem.

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Guided Practice: Example 2, continuedPredict the fuel efficiency at 20 mph and determine if the prediction seems to be reasonable. Substitute 20 into the equation and solve.

m(20) = –0.0146(20)2 + 1.1802(20) + 9.1356

m(20) ≈ 26.9

A fuel efficiency of 26.9 mpg at 20 mph is reasonable. This is an interpolation because it falls within the range of the given data. The given data goes from 18.6 to 62.1 mph and 20 falls within that range.

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Guided Practice: Example 2, continuedPredict the fuel efficiency at 65 mph and determine if the prediction seems to be reasonable.

Substitute 65 into the equation and solve.

m(65) = –0.0146(65)2 + 1.1802(65) + 9.1356

m(65) ≈ 24.2

A fuel efficiency of 24.2 mpg at 65 mph is reasonable. This is an extrapolation because it falls outside of the range of the given data. The given data goes from 18.6 to 62.1 mph and 65 falls outside of that range. However, it is close to the range of the given data, so the prediction is likely to be close. 38




m(75) = –0.0146(75)2 + 1.1802(75) + 9.1356

m(75) ≈ 15.5

A fuel efficiency of 15.5 mpg at 75 mph might not be reasonable. This is an extrapolation because it falls outside of the range of the given data. Points from outside of the data set may be more unreliable than those from within the data or close to the endpoints.

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m(90) = –0.0146(90)2 + 1.1802(90) + 9.1356

m(90) ≈ –2.9

A negative fuel efficiency is not only unreasonable, but also impossible. This is an extrapolation because it falls outside of the range of the given data. We must take care when extrapolating data to not use predictions that are unreasonable or impossible.

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Documents

Introduction Data surrounds us in the real world. Every day, people are presented with numbers and are expected to make predictions about future events