To determine whether the given table of values represents a linear, quadratic, or exponential function, we will analyze the pattern of the [tex]\( y \)[/tex]-values as [tex]\( x \)[/tex] increases. We do this by calculating the differences between successive [tex]\( y \)[/tex]-values:
Here is the given table of values:
[tex]\[
\begin{array}{|cc|}
\hline
x & y \\
\hline
-2 & 0 \\
-1 & 1.5 \\
0 & 3 \\
1 & 4.5 \\
2 & 6 \\
\hline
\end{array}
\][/tex]
Step 1: Calculate the first differences
We find the differences between each consecutive pair of [tex]\( y \)[/tex]-values:
[tex]\[
\begin{aligned}
y_{-1} - y_{-2} &= 1.5 - 0 = 1.5 \\
y_0 - y_{-1} &= 3 - 1.5 = 1.5 \\
y_1 - y_0 &= 4.5 - 3 = 1.5 \\
y_2 - y_1 &= 6 - 4.5 = 1.5 \\
\end{aligned}
\][/tex]
Step 2: Check for consistency in the differences
Notice that all the first differences are equal to 1.5:
[tex]\[
1.5, \; 1.5, \; 1.5, \; 1.5
\][/tex]
When the first differences are equal, the function is linear. This consistency in the differences indicates that the rate of change of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex] is constant.
Step 3: Conclude the function type
Since all the first differences are equal, the given table of values represents a linear function.
Therefore, the function given by the table of values is:
a. Linear
