Question

Is the function represented by the table linear?

x
y
10
–6
11
1
12
6
13
12
Yes, because it has a constant rate of change.
Yes, because it does not have a constant rate of change.
No, because it has a constant rate of change.
No, because it does not have a constant rate of change.

Answer 1

To determine if the function represented by the table is linear, we need to check if there is a constant rate of change between the values of \(y\) as \(x\) changes.

Let's calculate the differences:

1. From \(x = 10\) to \(x = 11\):
- \(y\) changes from \(-6\) to \(1\): \(1 - (-6) = 7\)
- Change in \(x\): \(11 - 10 = 1\)
- Rate of change: \(\frac{7}{1} = 7\)

2. From \(x = 11\) to \(x = 12\):
- \(y\) changes from \(1\) to \(6\): \(6 - 1 = 5\)
- Change in \(x\): \(12 - 11 = 1\)
- Rate of change: \(\frac{5}{1} = 5\)

3. From \(x = 12\) to \(x = 13\):
- \(y\) changes from \(6\) to \(12\): \(12 - 6 = 6\)
- Change in \(x\): \(13 - 12 = 1\)
- Rate of change: \(\frac{6}{1} = 6\)

Now we can summarize the rates of change:

- From \(x = 10\) to \(x = 11\): \(7\)
- From \(x = 11\) to \(x = 12\): \(5\)
- From \(x = 12\) to \(x = 13\): \(6\)

Since the rates of change are not constant (7, 5, and 6 are different), the function represented by the table does not have a linear relationship.

Therefore, the answer is:

**No, because it does not have a constant rate of change.**