The average retail price of a pair of designer shirts in Canada, from 1975 to 1999, can be
modelled by the polynomial function P(x) = 0.0045x
3 − 0.188x
2 + 4.331x + 18.65.
a. Describe the characteristics of the graph of the polynomial function. Include the
following:
• Degree of the function_____________
• Type of graph ___________________
• end behaviour___________________
• possible number of turning points _________________
b. Explain what the constant term means in the context of the problem.
c. What is the predicted value of shirts in 2003 using this function?
a.
- Degree of the function: 3
- Type of graph: Since the degree of the function is odd, the graph is a cubic function.
- End behavior: As x approaches negative infinity, the function approaches negative infinity. As x approaches positive infinity, the function approaches positive infinity.
- Possible number of turning points: The function has at most 2 turning points (points where the graph changes direction).
b. The constant term in the polynomial function represents the initial price of the shirts in 1975. In this case, the constant term of 18.65 represents the initial average retail price of a pair of designer shirts in Canada in 1975.
c. To find the predicted value of shirts in 2003, we need to plug in x = 28 (since 2003 is 28 years after 1975) into the function:
P(28) = 0.0045(28)^3 - 0.188(28)^2 + 4.331(28) + 18.65
P(28) = 0.0045(21952) - 0.188(784) + 121.368 + 18.65
P(28) = 98.784 - 147.392 + 121.368 + 18.65
P(28) = 91.41
Therefore, the predicted average retail price of a pair of designer shirts in Canada in 2003 is $91.41.