Question

Determine whether the relation defined by the table represents a linear function. Explain your answer.
Input
Output
2
15
5
negative 30
8
45
11
negative 60
14
75

Question content area bottom
Part 1
Does the relation represent a linear​ function?
A.
​No, because the rate of change is 5
B.
No comma because the rate of change is not constant.
C.
​Yes, because the rate of change is not constant.
D.
Yes comma because the rate of change is 5.

Answers

Answered by GPT-4o mini
To determine if the relation defined by the table represents a linear function, we need to check if the rate of change (slope) between the input (x) and output (y) values is constant.

Let's look at the given pairs of input and output:

1. For inputs 2 and 15:
- Change in x: \( 2 \rightarrow 5 \) is \( 5 - 2 = 3 \)
- Change in y: \( 15 \rightarrow -30 \) is \( -30 - 15 = -45 \)
- Rate of change (slope) = \( \frac{-45}{3} = -15 \)

2. For inputs 5 and -30:
- Change in x: \( 5 \rightarrow 8 \) is \( 8 - 5 = 3 \)
- Change in y: \( -30 \rightarrow 45 \) is \( 45 - (-30) = 75 \)
- Rate of change = \( \frac{75}{3} = 25 \)

3. For inputs 8 and 45:
- Change in x: \( 8 \rightarrow 11 \) is \( 11 - 8 = 3 \)
- Change in y: \( 45 \rightarrow -60 \) is \( -60 - 45 = -105 \)
- Rate of change = \( \frac{-105}{3} = -35 \)

4. For inputs 11 and -60:
- Change in x: \( 11 \rightarrow 14 \) is \( 14 - 11 = 3 \)
- Change in y: \( -60 \rightarrow 75 \) is \( 75 - (-60) = 135 \)
- Rate of change = \( \frac{135}{3} = 45 \)

Now, let's summarize the rates of change:
- From (2, 15) to (5, -30): -15
- From (5, -30) to (8, 45): 25
- From (8, 45) to (11, -60): -35
- From (11, -60) to (14, 75): 45

Since the rates of change are not constant (they vary between -15, 25, -35, and 45), the relation does not represent a linear function.

The correct answer is:
**B. No, because the rate of change is not constant.**

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