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Question 2 (Multiple Choice, Worth 2 points)
(Linear Functions MC)

Which statement best explains whether the following table represents a linear or a nonlinear function?

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
-2 & [tex]$\frac{1}{9}$[/tex] \\
\hline
-1 & [tex]$\frac{1}{3}$[/tex] \\
\hline
0 & 1 \\
\hline
1 & 3 \\
\hline
2 & 9 \\
\hline
\end{tabular}

A. The table represents a linear function because the rate of change is constant.

B. The table represents a linear function because the rate of change is not constant.

C. The table represents a nonlinear function because the rate of change is constant.

D. The table represents a nonlinear function because the rate of change is not constant.

Sagot :

GinnyAnsweridn
To determine whether the given table represents a linear or nonlinear function, we need to examine the relationship between the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values by calculating the rate of change, which is essentially the slope between consecutive points.

The table is as follows:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -2 & \frac{1}{9} \\ \hline -1 & \frac{1}{3} \\ \hline 0 & 1 \\ \hline 1 & 3 \\ \hline 2 & 9 \\ \hline \end{array} \][/tex]

To find the rate of change between each pair of points:
1. Calculate the change in [tex]\(y\)[/tex] values divided by the change in [tex]\(x\)[/tex] values for each pair of consecutive points.

[tex]\[ \Delta y = y_{n+1} - y_n \][/tex]
[tex]\[ \Delta x = x_{n+1} - x_n \][/tex]
[tex]\[ \text{rate of change} = \frac{\Delta y}{\Delta x} \][/tex]

Let's compute this step-by-step for each pair of points:

### Between [tex]\(x = -2\)[/tex] and [tex]\(x = -1\)[/tex]:
[tex]\[ \Delta y = \frac{1}{3} - \frac{1}{9} = \frac{3}{9} - \frac{1}{9} = \frac{2}{9} \][/tex]
[tex]\[ \Delta x = -1 - (-2) = 1 \][/tex]
[tex]\[ \text{rate of change} = \frac{2/9}{1} = \frac{2}{9} \][/tex]

### Between [tex]\(x = -1\)[/tex] and [tex]\(x = 0\)[/tex]:
[tex]\[ \Delta y = 1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3} \][/tex]
[tex]\[ \Delta x = 0 - (-1) = 1 \][/tex]
[tex]\[ \text{rate of change} = \frac{2/3}{1} = \frac{2}{3} \][/tex]

### Between [tex]\(x = 0\)[/tex] and [tex]\(x = 1\)[/tex]:
[tex]\[ \Delta y = 3 - 1 = 2 \][/tex]
[tex]\[ \Delta x = 1 - 0 = 1 \][/tex]
[tex]\[ \text{rate of change} = \frac{2}{1} = 2 \][/tex]

### Between [tex]\(x = 1\)[/tex] and [tex]\(x = 2\)[/tex]:
[tex]\[ \Delta y = 9 - 3 = 6 \][/tex]
[tex]\[ \Delta x = 2 - 1 = 1 \][/tex]
[tex]\[ \text{rate of change} = \frac{6}{1} = 6 \][/tex]

As seen from these calculations, the rate of change is not constant.

Since the rate of change is not constant, the given table does not represent a linear function. Therefore, the correct statement is:

The table represents a nonlinear function because the rate of change is not constant.
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