Question

Determine whether the relation defined by the table represents a linear function. Explain your answer.
Input
Output
4
22
7
negative 34
10
46
13
negative 58
16
70

Question content area bottom
Part 1
Does the relation represent a linear​ function?
A.
​No, because the rate of change is 4
B.
​Yes, because the rate of change is not constant.
C.
Yes comma because the rate of change is 4.
D.
No comma because the rate of change is not constant.

Answer 1

To determine whether the relation represents a linear function, we need to check if the changes in the outputs (y-values) for consistent changes in the inputs (x-values) create a constant rate of change.

Let's analyze the given inputs and outputs:

\[
\begin{array}{|c|c|}
\hline
\text{Input (x)} & \text{Output (y)} \\
\hline
4 & 22 \\
7 & -34 \\
10 & 46 \\
13 & -58 \\
16 & 70 \\
\hline
\end{array}
\]

Next, we will calculate the change in outputs corresponding to the change in inputs.

1. From **4 to 7** (change of 3):
- Output changes from 22 to -34: \( -34 - 22 = -56 \)
- Rate of change = \( \frac{-56}{3} = -18.67 \)

2. From **7 to 10** (change of 3):
- Output changes from -34 to 46: \( 46 - (-34) = 80 \)
- Rate of change = \( \frac{80}{3} = 26.67 \)

3. From **10 to 13** (change of 3):
- Output changes from 46 to -58: \( -58 - 46 = -104 \)
- Rate of change = \( \frac{-104}{3} = -34.67 \)

4. From **13 to 16** (change of 3):
- Output changes from -58 to 70: \( 70 - (-58) = 128 \)
- Rate of change = \( \frac{128}{3} = 42.67 \)

The rates of change calculated are not consistent, as they vary significantly between different segments:

- From 4 to 7: -18.67
- From 7 to 10: 26.67
- From 10 to 13: -34.67
- From 13 to 16: 42.67

Since the rate of change is not constant, we can conclude that the relation does not represent a linear function.

Based on this conclusion, the correct answer is:

**D. No, because the rate of change is not constant.**