To determine whether the given function is linear or exponential, we will analyze the differences between successive [tex]\( y \)[/tex]-values. Here's a step-by-step explanation:
1. Define the given data:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 26 \\
\hline
2 & 44 \\
\hline
3 & 62 \\
\hline
4 & 80 \\
\hline
5 & 98 \\
\hline
\end{array}
\][/tex]
2. Calculate the differences between successive [tex]\( y \)[/tex]-values:
[tex]\[
\begin{aligned}
\Delta y_1 &= y_2 - y_1 = 44 - 26 = 18, \\
\Delta y_2 &= y_3 - y_2 = 62 - 44 = 18, \\
\Delta y_3 &= y_4 - y_3 = 80 - 62 = 18, \\
\Delta y_4 &= y_5 - y_4 = 98 - 80 = 18.
\end{aligned}
\][/tex]
3. Analyze the differences:
We observe that the differences ([tex]\(\Delta y\)[/tex]) between each consecutive [tex]\( y \)[/tex]-value are all equal to 18. When the differences between consecutive [tex]\( y \)[/tex]-values are constant, the function is linear.
4. Compare with the given options:
- Option A: It is an exponential function because the factor between each [tex]\( x \)[/tex] and [tex]\( y \)[/tex]-value is constant.
- This is not correct because exponential functions have a constant ratio between successive [tex]\( y \)[/tex]-values, not constant differences.
- Option B: It is a linear function because the difference [tex]\( y - x \)[/tex] for each row is constant.
- There seems to be a slight error in the wording. This option could be more precisely stated as: "It is a linear function because the differences between consecutive [tex]\( y \)[/tex]-values are constant." However, given the context and the equal differences calculated, this is the most fitting option.
- Option C: It is an exponential function because the [tex]\( y \)[/tex]-values increase by an equal factor over equal intervals of [tex]\( x \)[/tex]-values.
- This is incorrect because an exponential function would increase by a multiplicative factor, not by a constant additive difference.
Thus, the correct statement about the function represented by the table is:
B. It is a linear function because the difference [tex]\( y-x \)[/tex] for each row is constant.
