a First, rewrite the values in the first row of the table, taking year 1981 as 0. Then, use a graphing calculator to find a quadratic regression.
B
b Recall that the price cannot be negative. Therefore, all values of the range must include only non-negative numbers. Also, note that the domain and range are discrete sets.
C
c Substitute y=37 into the equation found in Part A and solve for x.
D
d Substitute y=50 into the equation found in Part A and solve for x. Is the obtained value in the domain of the function?
A
aExample Model: y=-0.004x^2+0.93x+18.586
B
bExample Domain: Integers from -10 to 27. Example Range: Whole numbers from 9 to 42.
C
c Year 2003.
D
dPrediction: Year 2022. Explanation: The year is outside the domain of the data points.
Practice makes perfect
a We want to find a quadratic model for the data in the given table.
Price of First-Class Stamp
Year
1981
1991
1995
1999
2001
2006
2007
2008
Price (cents)
18
29
32
33
34
39
41
42
Let's consider year the 1981 as year 0. Therefore, 1991 will be year 10, 1995 will be year 14, and so on.
Price of First-Class Stamp
Year
0
10
14
18
20
25
26
27
Price (cents)
18
29
32
33
34
39
41
42
Using a graphing calculator, let's make a scatter plot of the years and prices. First we have to enter the values into lists by pushing STAT, choosing Edit, and then entering the years in the first column and the prices in the second column.
Having entered the values, we can plot them by pushing 2nd and Y=, and then choosing one of the plots in the list. Make sure to turn the plot ON, choose the scatter plot as the type, use L1 and L2 as XList and YList, and finally we can pick whatever mark we want.
To make a quadratic regression, we push STAT, scroll right to CALC, and then choose the fifth option in the list, which is QuadReg.
Now that we know the the values of a, b, and c, we can write the quadratic function that models the given data. Let's round the values to three decimal places.
y=-0.004x^2+0.93x+18.586
To graph this function, we press Y= and then write the quadratic equation. After this, we press GRAPH. Be aware that we may need to adjust the window to see the graph and the points.
b Since the domain represents the years, we will consider it as a discrete set. Therefore, a reasonable domain for the model could be the integers from x=-10, which represents the year 1971, to x=27, which represents the year 2008.
Domain: Integers from-10 to 27Let's find the range. To do so, note that the price is always increasing. Therefore, we can calculate the lower and upper boundaries of the range by substituting x=-10 and x=27, respectively, into the equation found in Part A. Let's start with x=- 10.
From the table, we know that when x=27 the y-value must be close to 42. With this information, we can set the range for our model.
Domain:& Integers from-10 to 27
Range:& Whole numbers from 9 to 42
Notice that the data we found is just a sample. We could have chosen a different reasonable domain and range.
c To determine when the price of the first-class postage was 37 cents, we must solve the equation 37=-0.004x^2+0.93x+18.586. First, we simplify the equation.
By using the positive and the negative sign we get two different answers.
x_(1,2)=0.93± 0.755166.../0.008
x_1=0.93+0.755166.../0.008
x_2=0.93-0.755166.../0.008
x_1 ≈ 210.6
x_2≈ 21.9
Since the domain of our function is discrete we can say that x_2=22, which corresponds to the year 1981+22=2003. Therefore, the price was 37 cents in 2003. Similarly, we can say that x_1=211, which corresponds to the year 1981+221=2202. This is the year in which the price will be 37 cents again.
d As in Part C, to know when the price will be 50 cents we have to solve the equation 50=-0.004x^2+0.93x+18.586. First, we simplify the equation.
By using the positive and the negative sign, we get two different answers.
x_(1,2)=0.93± 0.601893.../0.008
x_1=0.93+0.601893.../0.008
x_2=0.93-0.601893.../0.008
x_1 ≈ 191.5
x_2 ≈ 41
Notice that x_2≈ 41 corresponds to the year 2022 and x_1≈ 191.5 corresponds to year 2173. Since both values are outside the domain of our function, these solutions may not be valid.
