Question

A function is represented by the values in the table.

x y
22 26
20 22
16 20
14 18
10 14


The function represented in the table ______ linear.

Answer 1

To determine if the function represented by the values in the table is linear, we need to check if the changes in \(y\) are consistent with the changes in \(x\). We can do this by calculating the differences in \(y\) for the corresponding differences in \(x\).

Let's analyze the function step by step:

1. Calculate the differences in \(x\) and \(y\):
- From \(x = 22\) to \(x = 20\):
- \(y\) changes from \(26\) to \(22\)
- Change in \(x\): \(20 - 22 = -2\)
- Change in \(y\): \(22 - 26 = -4\)

- From \(x = 20\) to \(x = 16\):
- \(y\) changes from \(22\) to \(20\)
- Change in \(x\): \(16 - 20 = -4\)
- Change in \(y\): \(20 - 22 = -2\)

- From \(x = 16\) to \(x = 14\):
- \(y\) changes from \(20\) to \(18\)
- Change in \(x\): \(14 - 16 = -2\)
- Change in \(y\): \(18 - 20 = -2\)

- From \(x = 14\) to \(x = 10\):
- \(y\) changes from \(18\) to \(14\)
- Change in \(x\): \(10 - 14 = -4\)
- Change in \(y\): \(14 - 18 = -4\)

2. Now, let's summarize the changes:
- Change from \(22\) to \(20\): \( \Delta x = -2, \Delta y = -4 \)
- Change from \(20\) to \(16\): \( \Delta x = -4, \Delta y = -2 \)
- Change from \(16\) to \(14\): \( \Delta x = -2, \Delta y = -2 \)
- Change from \(14\) to \(10\): \( \Delta x = -4, \Delta y = -4 \)

Seeing the ratios of \(\Delta y\) to \(\Delta x\) (essentially the slope) is inconsistent across different intervals:

- First interval slope: \(-4 / -2 = 2\)
- Second interval slope: \(-2 / -4 = 0.5\)
- Third interval slope: \(-2 / -2 = 1\)
- Fourth interval slope: \(-4 / -4 = 1\)

Since the slope is not constant, the function represented in the table is **not linear**.

So the function represented in the table **is not** linear.