To determine the [tex]\( y \)[/tex]-coordinate of the [tex]\( y \)[/tex]-intercept of the graph of [tex]\( y = f(x) \)[/tex], we need to follow these steps:
1. Identify the two given points: From the table, we know that the points are [tex]\((-1, -2)\)[/tex] and [tex]\( (2, 4) \)[/tex].
2. Calculate the slope ([tex]\( m \)[/tex]) of the linear function:
- The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, f_1)\)[/tex] and [tex]\((x_2, f_2)\)[/tex] is:
[tex]\[
m = \frac{f_2 - f_1}{x_2 - x_1}
\][/tex]
- Substituting the given points into the formula:
[tex]\[
m = \frac{4 - (-2)}{2 - (-1)} = \frac{4 + 2}{2 + 1} = \frac{6}{3} = 2
\][/tex]
3. Write the equation of the line in the form [tex]\( f(x) = mx + b \)[/tex]:
- We know the slope [tex]\( m = 2 \)[/tex] and we need to find the [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex].
4. Use one of the given points to solve for the [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex]:
- Using the point [tex]\((x_1, f_1) = (-1, -2)\)[/tex], substitute into the equation [tex]\( f(x) = mx + b \)[/tex]:
[tex]\[
-2 = 2(-1) + b
\][/tex]
- Solve for [tex]\( b \)[/tex]:
[tex]\[
-2 = -2 + b \implies b = 0
\][/tex]
The [tex]\( y \)[/tex]-coordinate of the [tex]\( y \)[/tex]-intercept is [tex]\(\boxed{0}\)[/tex].
