First, we need to understand what the x-intercept of a graph is. An x-intercept is the point where the graph of a function crosses the x-axis. At this point, the value of [tex]\( y \)[/tex] (or [tex]\( f(x) \)[/tex]) is 0. Therefore, we need to find the value of [tex]\( x \)[/tex] that makes [tex]\( f(x) = 0 \)[/tex].
Given the data points in the table:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & -11 & -10 & -9 & -8 \\
\hline
f(x) & 21 & 18 & 15 & 12 \\
\hline
\end{array}
\][/tex]
First, let's represent the linear function [tex]\( f(x) \)[/tex] in the form of a linear equation [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the y-intercept.
Step 1: Calculate the slope (m) of the line using any two points from the given data:
Let's use the points (-11, 21) and (-10, 18).
[tex]\[
m = \frac{y2 - y1}{x2 - x1} = \frac{18 - 21}{-10 - (-11)} = \frac{-3}{1} = -3
\][/tex]
Step 2: Determine the y-intercept (b):
We can use the slope and one of the points to find the y-intercept. Let's use the point (-11, 21).
[tex]\[
21 = -3(-11) + b \implies 21 = 33 + b \implies b = 21 - 33 \implies b = -12
\][/tex]
Thus, the equation of the line is:
[tex]\[
y = -3x - 12
\][/tex]
Step 3: Find the x-intercept:
To find the x-intercept, we set [tex]\( y \)[/tex] to 0 and solve for [tex]\( x \)[/tex].
[tex]\[
0 = -3x - 12 \implies 3x = -12 \implies x = -4
\][/tex]
So, the x-intercept is [tex]\( (-4, 0) \)[/tex].
Matching this with the given options:
A. [tex]\( (-3, 0) \)[/tex]
B. [tex]\( (-4, 0) \)[/tex]
C. [tex]\( (-9, 0) \)[/tex]
D. [tex]\( (-12, 0) \)[/tex]
The correct answer is:
[tex]\[
\boxed{B. (-4, 0)}
\][/tex]
