To determine whether a table represents a linear function, we need to see if the changes in \( y \) values are consistent when the changes in \( x \) values are the same. In other words, a linear function will have constant differences between consecutive \( y \) values when \( x \) values increase by a constant amount.
Let's analyze the given table:
[tex]\[
\begin{tabular}{|c|c|}
\hline [tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline 1 & 5 \\
\hline 2 & 10 \\
\hline 3 & 15 \\
\hline 4 & 20 \\
\hline 5 & 25 \\
\hline
\end{tabular}
\][/tex]
We start by calculating the differences between consecutive \( y \) values:
- Difference between \( y_2 \) and \( y_1 \) is \( 10 - 5 = 5 \)
- Difference between \( y_3 \) and \( y_2 \) is \( 15 - 10 = 5 \)
- Difference between \( y_4 \) and \( y_3 \) is \( 20 - 15 = 5 \)
- Difference between \( y_5 \) and \( y_4 \) is \( 25 - 20 = 5 \)
We observe that the differences between consecutive \( y \) values are all \( 5 \). This consistency in differences indicates that the function is linear.
To further confirm, let's consider the general form of a linear function, which is \( y = mx + b \). Here,
- \( m \) is the slope (rate of change)
- \( b \) is the y-intercept (the value of \( y \) when \( x = 0 \))
Given the differences between consecutive \( y \) values being constant, we can deduce that the rate of change (\( m \)) is 5. The relationships between the \( x \) and \( y \) values can then be inferred to fit a linear equation of the form:
[tex]\[ y = 5x \][/tex]
Hence, the table given represents a linear function.
