a Look at the given table and graph, and think about the values belonging to the domain. How do you think the domain will be restricted?
B
b Look at the model's representations and your results from Part A. What can you conclude?
A
a
Example Domain
Example Range
Dry Roadway Model
[0,120]
[0,605]
Wet Roadway Model
[0,120]
[0,923]
B
b See solution.
Practice makes perfect
a We will start by reviewing the concepts of domain and range. After that we will find the domain. Then, we will explain how to find the function rules for both models to find the range.
Domain and Range
Let's review what the domain and range of a function mean.
The domain is the set of all x-values or inputs of a function.
The range is the set of all y-values or outputs of a function.
From the table and graph, we can see that our domain is comprised of the possible speeds of a certain automobile, and that the range will be the stopping distance corresponding to each speed.
Finding a Reasonable Domain
Since the average top speed of a car is around 120 mph and because we cannot have negative speeds, a reasonable domain would be [0,120].
Domain (Both Models) [0.8em]
[0,120]
On the other hand, to find the range for each case we need to have an idea of how each function assigns the y-values.
Finding a Reasonable Range
Notice that the values for the variable x are speeds and this is a continuous quantity. Therefore, the range can be obtained as the interval within the corresponding stopping distances for the minimum and maximum values of the domain interval endpoints.
Domain: & [ 0, 120]
Range: & [f( 0),f( 120)]
To do this we need to find a function rule for each model.
Finding the Functions Rules
Let's start with the table of values. We can plot these values to get an idea of the behavior of the function.
Notice that the points tend to follow a curved trajectory more than a straight line. For this reason, we can try a quadratic regression. We can do this using a graphic calculator. We will show the steps to do it.
Step 1: Press STAT and choose Edit. Then, enter the values into the lists.
Step 2: Press STAT again and scroll to right to view the CALC options, and then choose the fifth option in the list, QuadReg.
Step 3: We graph the data and the function to see how reasonable our model is.
In this case, the found quadratic equation fits the data very well. Therefore, the quadratic model is quite reasonable. Rounding the parameters a, b, and c to 3 decimal places, we obtain a quadratic function for the data in the table.
y =0.042x^2-0.041x+0.889
For the other case, we have a graph of a function which also suggests a non-linear behavior. We can try to localize some points approximately and repeat the process.
Let's use the points (0,0), (20,23), (40,66), and (50,100) to get a quadratic regression equation. Following the same steps as before, we obtain the equation below.
y=0.076x^2−1.479x+6.139
We graph the data points and the equation together to see how well the equation fits the data points.
We see that this time our quadratic model is not as precise as in the previous case. This can be because the function is not a quadratic relation, or due to the error when approximating the points location. However, we can use it as an approximation.
Using the Function Rules to Find the Range
Now that we have function rules for each relation, we can find the range.
Since both graphs start at 0, we just need to evaluate the functions at x=120.
Dry Roadway Model
y =0.042x^2-0.041x+0.889 [0.5em] [-0.6em]
Wet Roadway Model
y=0.076x^2−1.479x+6.139
Let's first evaluate the equation for the dry roadway when x=120.
We get about 605 feet for the stopping distance when driving at 120 mph on a dry roadway. We will do the same for the equation of the wet roadway stopping distance.
We find that the stopping distance for the wet roadway when going at 120 mph is about 923feet. Now we can define a reasonable range for both models.
Domain
Range
Dry Roadway Model
[0,120]
[0,605]
Wet Roadway Model
[0,120]
[0,923]
Notice that the criteria we used when defining a reasonable domain can change. We could have done it according to the speed limit allowed, for example. Furthermore, when finding the function rules we used approximations. Therefore, these are just examples solutions, and there could be other similar and valid answers.
b As we can see from the ranges, the stopping distance can be dramatically affected by the road conditions. The wet roadway stopping distances are noticeable greater than those for a dry roadway.
