Question

for this item, select the answers in the choice matrix by clicking the appropriate boxes in each row. determine whether each table represents a linear or exponential function. exponential linear \

$$\begin{tabular}{|c|c|} \\hline \\( x \\) & \\( y \\) \\\\ \\hline -2 & -8 \\\\ \\hline 0 & -12 \\\\ \\hline 3 & -18 \\\\ \\hline 4 & -20 \\\\ \\hline \\end{tabular}$$

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$$\begin{tabular}{|c|c|} \\hline \\( x \\) & \\( y \\) \\\\ \\hline -2 & 9 \\\\ \\hline 0 & 36 \\\\ \\hline 3 & 288 \\\\ \\hline 4 & 576 \\\\ \\hline \\end{tabular}$$

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$$\begin{tabular}{|c|c|} \\hline \\( x \\) & \\( y \\) \\\\ \\hline -2 & 64 \\\\ \\hline 0 & 16 \\\\ \\hline 3 & 2 \\\\ \\hline 4 & 1 \\\\ \\hline \\end{tabular}$$

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$$\begin{tabular}{|c|c|} \\hline \\( x \\) & \\( y \\) \\\\ \\hline -2 & 1.5 \\\\ \\hline 0 & 2.5 \\\\ \\hline 3 & 4 \\\\ \\hline 4 & 4.5 \\\\ \\hline \\end{tabular}$$

Explanation:

Step1: Test first table for linearity

Check constant $\Delta y$:
$\Delta y_1 = -12 - (-8) = -4$
$\Delta y_2 = -18 - (-12) = -6$
$\Delta y_3 = -20 - (-18) = -2$
Wait, recalculate slope between points:
Slope $m_1 = \frac{-12 - (-8)}{0 - (-2)} = \frac{-4}{2} = -2$
Slope $m_2 = \frac{-18 - (-12)}{3 - 0} = \frac{-6}{3} = -2$
Slope $m_3 = \frac{-20 - (-18)}{4 - 3} = \frac{-2}{1} = -2$
Constant slope confirms linear.

Step2: Test second table for exponentiality

Check constant ratio of $y$:
$\frac{36}{9} = 4$
$\frac{288}{36} = 8$
Wait, check ratio for equal $\Delta x$:
From $x=-2$ to $x=0$ ($\Delta x=2$), ratio $4=2^2$; from $x=0$ to $x=3$ ($\Delta x=3$), ratio $8=2^3$; from $x=3$ to $x=4$ ($\Delta x=1$), ratio $\frac{576}{288}=2=2^1$.
Constant base ratio of 2 per $\Delta x=1$, so exponential.

Step3: Test third table for exponentiality

Check constant ratio of $y$:
$\frac{16}{64} = \frac{1}{4}$
$\frac{2}{16} = \frac{1}{8}$
Wait, for $\Delta x=2$ (x=-2 to 0): ratio $\frac{1}{4}=(\frac{1}{2})^2$; $\Delta x=3$ (x=0 to 3): ratio $\frac{2}{16}=\frac{1}{8}=(\frac{1}{2})^3$; $\Delta x=1$ (x=3 to 4): ratio $\frac{1}{2}=(\frac{1}{2})^1$.
Constant base ratio of $\frac{1}{2}$ per $\Delta x=1$, so exponential.

Step4: Test fourth table for linearity

Check constant slope:
Slope $m_1 = \frac{2.5 - 1.5}{0 - (-2)} = \frac{1}{2} = 0.5$
Slope $m_2 = \frac{4 - 2.5}{3 - 0} = \frac{1.5}{3} = 0.5$
Slope $m_3 = \frac{4.5 - 4}{4 - 3} = \frac{0.5}{1} = 0.5$
Constant slope confirms linear.

Answer:

1. First table: Linear
2. Second table: Exponential
3. Third table: Exponential
4. Fourth table: Linear