Question

the data table contains a portion of data points from a function for average daily high temperatures throughout the year in a certain city.
a. identify the independent and dependent variables, and describe the domain and the range.
b. make a clear graph of the function.
c. describe the function in words.
jan. 1
feb. 1
mar.1
apr. 1
may 1
june 1
july 1
aug. 1
sep. 1
oct. 1
nov. 1
dec. 1
dec. 31
42°f
38°f
49°f
57°f
67°f
77°f
87°f
83°f
83°f
67°f
52°f
48°f
44°f
b. choose the correct graph.
oa.
ob.
oc.
od.
e high temperature
e high temperature
e high temperature
e high temperature

Explanation:

Part a

Step1: Identify Variables

Independent: Time (months/days of the year), as it's the input. Dependent: Average daily high temperature, as it depends on time.

Step2: Domain Description

Domain: All days/months in the year (from Jan 1 to Dec 31, represented by the given dates), so the set of input time values.

Step3: Range Description

Range: The set of temperature values. From the data, the lowest is \(38^\circ\text{F}\) (Feb 1), highest is \(87^\circ\text{F}\) (July 1), so range is \(38^\circ\text{F} \leq T \leq 87^\circ\text{F}\) (where \(T\) is temperature).

Step1: Analyze Data Trend

• Jan 1: \(42^\circ\text{F}\), Feb 1: \(38^\circ\text{F}\) (slight drop)
• Feb 1 to July 1: Temperatures increase (38, 49, 57, 67, 77, 87)
• July 1 to Aug 1: Slight drop (87 to 83)
• Aug 1 to Sep 1: Stable (83)
• Sep 1 to Oct 1: Drop (83 to 67)
• Oct 1 to Nov 1: Drop (67 to 52)
• Nov 1 to Dec 1: Drop (52 to 48)
• Dec 1 to Dec 31: Slight increase (48 to 44? Wait, 48 to 44 is a drop? Wait, 48 to 44 is a decrease, then maybe a small rise? Wait, data: Dec 1 is 48, Dec 31 is 44? Wait, maybe typo, but following data: after July, temp decreases with some stability.

Step2: Match with Graphs

• Graph A: Shows a rise to July, then decline, with the shape matching the temperature trend (drop in Feb, rise to July, then decline, stable in Aug-Sep, then decline).
• Graph B: Has erratic peaks, not matching.
• Graph C: Starts high and declines then rises, opposite of data (data starts at 42, drops to 38, then rises).
• Graph D: Flat with small changes, not matching the rise to July.

Step1: Describe Trend

The function (average daily high temperature over the year) starts at \(42^\circ\text{F}\) on Jan 1, drops slightly to \(38^\circ\text{F}\) on Feb 1, then increases steadily from Feb 1 to July 1 (reaching \(87^\circ\text{F}\)). After July 1, the temperature decreases slightly to \(83^\circ\text{F}\) by Aug 1, remains stable at \(83^\circ\text{F}\) until Sep 1, then decreases steadily from Sep 1 to Nov 1 (reaching \(52^\circ\text{F}\)), continues decreasing to \(48^\circ\text{F}\) by Dec 1, and has a small decrease (or slight change) to \(44^\circ\text{F}\) by Dec 31. In summary, the temperature has a winter dip in February, rises to a summer peak in July, then declines through fall and winter, with a brief stability in late summer.

Answer:

Independent variable: Time (months/days of the year). Dependent variable: Average daily high temperature. Domain: Time points (Jan 1, Feb 1, ..., Dec 31). Range: Temperatures from \(38^\circ\text{F}\) to \(87^\circ\text{F}\) (inclusive of all values in between from the data).

Part b