To find the rate of change for the given data points, we need to determine the change in [tex]\( y \)[/tex] values over the change in [tex]\( x \)[/tex] values from the first data point to the last.
Here's the data provided in the table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
x & y \\
\hline
-1 & 7 \\
\hline
2 & -2 \\
\hline
7 & -17 \\
\hline
10 & -26 \\
\hline
\end{tabular}
\][/tex]
To calculate the rate of change between the first and last points, we use the formula:
[tex]\[
\text{Rate of change} = \frac{\text{change in } y}{\text{change in } x}
\][/tex]
Using the first point [tex]\((-1, 7)\)[/tex] and the last point [tex]\((10, -26)\)[/tex]:
1. Identify the [tex]\( y \)[/tex]-values and [tex]\( x \)[/tex]-values:
- Initial [tex]\( y \)[/tex]-value, [tex]\( y_i = 7 \)[/tex]
- Final [tex]\( y \)[/tex]-value, [tex]\( y_f = -26 \)[/tex]
- Initial [tex]\( x \)[/tex]-value, [tex]\( x_i = -1 \)[/tex]
- Final [tex]\( x \)[/tex]-value, [tex]\( x_f = 10 \)[/tex]
2. Calculate the change in [tex]\( y \)[/tex]:
[tex]\[
\Delta y = y_f - y_i = -26 - 7 = -33
\][/tex]
3. Calculate the change in [tex]\( x \)[/tex]:
[tex]\[
\Delta x = x_f - x_i = 10 - (-1) = 10 + 1 = 11
\][/tex]
4. Calculate the rate of change:
[tex]\[
\text{Rate of change} = \frac{\Delta y}{\Delta x} = \frac{-33}{11} = -3
\][/tex]
Therefore, the rate of change for the given data is [tex]\(-3\)[/tex]. This means for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 3 units. Hence, the correct answer is:
[tex]\[
-3
\][/tex]
