Question

is the function represented by the table non-linear?

\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline 6 & 4 \\\\ \\hline 7 & 2 \\\\ \\hline 8 & 0 \\\\ \\hline 9 & -2 \\\\ \\hline \\end{array}$$

• 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.

Explanation:

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<reasoning>

Calculate the rate of change between consecutive points

\[

$$\begin{aligned} m_1 &= \frac{2 - 4}{7 - 6} = \frac{-2}{1} = -2 \\ m_2 &= \frac{0 - 2}{8 - 7} = \frac{-2}{1} = -2 \\ m_3 &= \frac{-2 - 0}{9 - 8} = \frac{-2}{1} = -2 \end{aligned}$$

\]

Determine if the function is linear

\[
\text{Since } m_1 = m_2 = m_3 = -2, \text{ the rate of change is constant.}
\]
\[
\text{A constant rate of change means the function is linear.}
\]
\[
\text{Therefore, the function is NOT non-linear.}
\]
</reasoning>

<answer>
<mcq-option>Yes, because it has a constant rate of change.</mcq-option>
<mcq-option>Yes, because it does not have a constant rate of change.</mcq-option>
<mcq-correct>No, because it has a constant rate of change.</mcq-correct>
<mcq-option>No, because it does not have a constant rate of change.</mcq-option>
</answer>

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Answer:

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<reasoning>

Calculate the rate of change between consecutive points

\[

$$\begin{aligned} m_1 &= \frac{2 - 4}{7 - 6} = \frac{-2}{1} = -2 \\ m_2 &= \frac{0 - 2}{8 - 7} = \frac{-2}{1} = -2 \\ m_3 &= \frac{-2 - 0}{9 - 8} = \frac{-2}{1} = -2 \end{aligned}$$

\]

Determine if the function is linear

\[
\text{Since } m_1 = m_2 = m_3 = -2, \text{ the rate of change is constant.}
\]
\[
\text{A constant rate of change means the function is linear.}
\]
\[
\text{Therefore, the function is NOT non-linear.}
\]
</reasoning>

<answer>
<mcq-option>Yes, because it has a constant rate of change.</mcq-option>
<mcq-option>Yes, because it does not have a constant rate of change.</mcq-option>
<mcq-correct>No, because it has a constant rate of change.</mcq-correct>
<mcq-option>No, because it does not have a constant rate of change.</mcq-option>
</answer>

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