To find the value of [tex]\( f(-6) \)[/tex] for the linear function [tex]\( f(x) \)[/tex], given the values in the table, we need to follow these steps:
1. Identify the points given in the table:
[tex]\[
(0, -1), (3, -2), (12, -5)
\][/tex]
2. Determine the slope [tex]\( m \)[/tex] of the linear function:
The slope [tex]\( m \)[/tex] can be determined by using any two points from the table. Let's use the points [tex]\((0, -1)\)[/tex] and [tex]\((3, -2)\)[/tex].
The formula for the slope [tex]\( m \)[/tex] is:
[tex]\[
m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
Plugging in the values from the points [tex]\((0, -1)\)[/tex] and [tex]\((3, -2)\)[/tex]:
[tex]\[
m = \frac{-2 - (-1)}{3 - 0} = \frac{-2 + 1}{3} = \frac{-1}{3} = -\frac{1}{3}
\][/tex]
3. Determine the y-intercept [tex]\( b \)[/tex]:
Since the point [tex]\((0, -1)\)[/tex] is given, it directly helps us find the y-intercept [tex]\( b \)[/tex]:
[tex]\[
f(0) = b = -1
\][/tex]
4. Formulate the linear function [tex]\( f(x) \)[/tex]:
With the slope [tex]\( m \)[/tex] and y-intercept [tex]\( b \)[/tex], the linear function [tex]\( f(x) \)[/tex] can be written as:
[tex]\[
f(x) = -\frac{1}{3}x - 1
\][/tex]
5. Calculate [tex]\( f(-6) \)[/tex]:
Substitute [tex]\( x = -6 \)[/tex] into the linear function:
[tex]\[
f(-6) = -\frac{1}{3}(-6) - 1
\][/tex]
Calculate the value:
[tex]\[
f(-6) = \frac{6}{3} - 1 = 2 - 1 = 1
\][/tex]
Therefore, the value of [tex]\( f(-6) \)[/tex] is [tex]\(\boxed{1}\)[/tex].
