To determine the equation of the form [tex]\( y = mx + b \)[/tex] from the given table of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values, follow these steps:
1. Identify the given [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & -2 & -1 & 0 & 1 & 2 \\
\hline
y & -2 & 0 & 2 & 4 & 6 \\
\hline
\end{array}
\][/tex]
2. Check if the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is linear:
- Calculate the differences between consecutive [tex]\( y \)[/tex] values:
[tex]\[
\begin{align*}
y_1 - y_0 &= 0 - (-2) = 2 \\
y_2 - y_1 &= 2 - 0 = 2 \\
y_3 - y_2 &= 4 - 2 = 2 \\
y_4 - y_3 &= 6 - 4 = 2 \\
\end{align*}
\][/tex]
- The differences are constant ([tex]\( 2, 2, 2, 2 \)[/tex]), indicating a linear relationship.
3. Find the slope [tex]\( m \)[/tex]:
- Since the differences in [tex]\( y \)[/tex] are consistent, the slope [tex]\( m \)[/tex] (also the rate of change of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex]) can be directly observed:
[tex]\[
m = 2
\][/tex]
4. Determine the y-intercept [tex]\( b \)[/tex]:
- Use the point where [tex]\( x = 0 \)[/tex], which is [tex]\((0, 2)\)[/tex]:
[tex]\[
b = 2
\][/tex]
5. Form the equation:
- Substitute the slope [tex]\( m \)[/tex] and y-intercept [tex]\( b \)[/tex] into the equation [tex]\( y = mx + b \)[/tex]:
[tex]\[
y = 2x + 2
\][/tex]
Thus, the equation that represents the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] from the given table is:
[tex]\[
y = 2x + 2
\][/tex]
