Question

analyzing a table to determine the type of function
this table shows data collected by a runner.

time (minutes)	1	2	3	4	5	6
elevation (meters)	12	26	67	98	124	145

which statement about the scenario represented in the table is true? assume time is the independent variable.

• the distance run is a nonlinear function because it does not have a constant rate of change.
• the elevation is a nonlinear function because it does not have a constant rate of change.
• both the distance run and the elevation are nonlinear functions because they do not have constant rates of change.
• both the distance run and the elevation are linear functions because they have a constant rate of change.

Explanation:

To solve this, we analyze the rate of change for distance and elevation with respect to time.

Step 1: Analyze Distance (miles) over Time (minutes)

The time increases by 1 minute each time (from 1 to 2, 2 to 3, etc.).

• From \( t = 1 \) to \( t = 2 \): \( \Delta \text{Distance} = 0.38 - 0.19 = 0.19 \) miles
• From \( t = 2 \) to \( t = 3 \): \( \Delta \text{Distance} = 0.57 - 0.38 = 0.19 \) miles
• From \( t = 3 \) to \( t = 4 \): \( \Delta \text{Distance} = 0.76 - 0.57 = 0.19 \) miles
• From \( t = 4 \) to \( t = 5 \): \( \Delta \text{Distance} = 0.95 - 0.76 = 0.19 \) miles
• From \( t = 5 \) to \( t = 6 \): \( \Delta \text{Distance} = 1.14 - 0.95 = 0.19 \) miles

The distance has a constant rate of change (0.19 miles per minute), so distance is a linear function.

Step 2: Analyze Elevation (meters) over Time (minutes)

Time increases by 1 minute each time.

• From \( t = 1 \) to \( t = 2 \): \( \Delta \text{Elevation} = 26 - 12 = 14 \) meters
• From \( t = 2 \) to \( t = 3 \): \( \Delta \text{Elevation} = 67 - 26 = 41 \) meters
• From \( t = 3 \) to \( t = 4 \): \( \Delta \text{Elevation} = 98 - 67 = 31 \) meters
• From \( t = 4 \) to \( t = 5 \): \( \Delta \text{Elevation} = 124 - 98 = 26 \) meters
• From \( t = 5 \) to \( t = 6 \): \( \Delta \text{Elevation} = 145 - 124 = 21 \) meters

The elevation has a non - constant rate of change (the differences are 14, 41, 31, 26, 21—no consistent value), so elevation is a nonlinear function.

Step 3: Evaluate the Statements

• "The distance run is a nonlinear function because it does not have a constant rate of change." → False (distance has a constant rate of change).
• "The elevation is a nonlinear function because it does not have a constant rate of change." → True (elevation’s rate of change is not constant).
• "Both the distance run and the elevation are nonlinear functions because they do not have constant rates of change." → False (distance has a constant rate of change).
• "Both the distance run and the elevation are linear functions because they have a constant rate of change." → False (elevation does not have a constant rate of change).

Answer:

The elevation is a nonlinear function because it does not have a constant rate of change.