Unit 5: Quadratic Functions
Fitting Quadratic Functions to Data
Solving Problems Given Functions Fitted to Data
1. The 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. A scatter plot of the data
with the graph of this model is shown below.
Use 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.
2. Use 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.
The 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 below.
3. The table below shows the height in feet of a children’s roller coaster at different times
throughout the ride.
Create a scatter plot of the data. Should a quadratic regression model be used to model the height of
the roller coaster? If so, find a quadratic equation that fits this data.
4. Look at the data given in the table that follows. What is the most appropriate regression for
the data: linear, exponential, or quadratic? Create a scatter plot of the data and confirm the
appropriateness of the model chosen.
Fitting a Function to Data
1. The students in Ms. Swan’s class surveyed people of all ages to find out how many people in
each of the age groups below exercise on a regular basis. Use your calculator to make a scatter
plot of the data in the table and to find a quadratic regression of this data. Use the “Group
number” column in the table to represent the age group, the x-values. Graph the regression
equation on top of your scatter plot.
2. The number of calories recommended for a healthy diet varies depending on your age and
level of activity. The table below shows the recommended number of calories required for
moderately active females at various ages. Use your calculator to make a scatter plot of this
data and to find a quadratic regression model for this situation. Then evaluate the strengths
and weaknesses of the regression model.
3. Doctors recommend that most people exercise for 30 minutes every day to stay healthy. To get
the best results, a person’s heart rate while exercising should reach between 50% and 75% of
his or her maximum heart rate, which is usually found by subtracting your age from 220. The
peak rate should occur at around the 25th minute of exercise. Alice is 30 years old, and her
maximum heart rate is 190 beats per minute (bpm). Assume that her resting rate is 60 bpm.
The table below shows Alice’s heart rate as it is measured every 5 minutes for 30 minutes
while she exercises. Interpret the model.
Make a scatter plot of the data. Use a graphing calculator to find a quadratic regression model
for the data. Use your model to extrapolate Alice’s heart rate after 35 minutes of exercise.