Question

at sunrise, the temperature is 55 °f. it rises steadily until noon. from noon until about 3:00 p.m., the temperature remains about the same. it slowly cools off after 3:00 p.m. then the temperature falls quickly after sunset. what does the graph of this function look like?

1. identify the input and output.
2. sketch the graph of the function.
3. how would you describe the graph of the function in the second interval?

on the back!
temperature
time

Explanation:

1. Identify the input and output.

Step1: Determine Input (Independent Variable)

The independent variable here is time (measured in hours from sunrise, or specific time intervals like sunrise to noon, noon to 3 PM, etc.). It's the variable we can control or that progresses on its own.

Step2: Determine Output (Dependent Variable)

The dependent variable is temperature (in °F), as it depends on the time of day. The temperature changes as time passes, so it's the output of the function where time is the input.

Step1: Define Axes

• x - axis (Input): Time (label intervals: sunrise, noon, 3 PM, sunset).
• y - axis (Output): Temperature (start at 55 °F, with a reasonable scale for temperature rise, plateau, and fall).

Step2: Plot Intervals

• Sunrise to Noon: Temperature rises steadily. Draw a line segment with a positive slope (from (sunrise, 55) to (noon, \( T_{\text{noon}} \)) where \( T_{\text{noon}} > 55 \)).
• Noon to 3 PM: Temperature remains the same (plateau). Draw a horizontal line segment (from (noon, \( T_{\text{noon}} \)) to (3 PM, \( T_{\text{noon}} \))).
• 3 PM to Sunset: Temperature falls quickly. Draw a line segment with a steeper negative slope (from (3 PM, \( T_{\text{noon}} \)) to (sunset, \( T_{\text{sunset}} \)) where \( T_{\text{sunset}} < T_{\text{noon}} \)).

(Note: The sketch should visually represent these three phases: increasing, constant, decreasing, with the last decrease being steeper than the initial increase.)

Step1: Identify the Second Interval

The second interval is from noon to about 3 PM.

Step2: Analyze the Graph in This Interval

During this time, the temperature remains about the same (constant). On a graph, a constant function (where output doesn’t change with input) is represented by a horizontal line segment (slope = 0).

Answer:

Input: Time (e.g., hours since sunrise, or time intervals like sunrise - noon, noon - 3 PM, 3 PM - sunset)
Output: Temperature (in °F)

2. Sketch the graph of the function.