To determine the linear relationship between [tex]\( x \)[/tex] and [tex]\( f(x) \)[/tex] as shown in the table, we need to find the slope ([tex]\( m \)[/tex]) and the y-intercept ([tex]\( b \)[/tex]) of the line.
Given data points are:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-1 & -7 \\
\hline
0 & -5 \\
\hline
1 & -3 \\
\hline
2 & -1 \\
\hline
3 & 1 \\
\hline
\end{array}
\][/tex]
### Step 1: Determine the slope ([tex]\( m \)[/tex])
1. We will use the slope formula that uses any two points on the line:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
2. Choose two points from the table to calculate the slope. Here we will use the points (0, -5) and (1, -3):
Using the points [tex]\((0, -5)\)[/tex] and [tex]\((1, -3)\)[/tex], we plug them into the formula:
[tex]\[
m = \frac{-3 - (-5)}{1 - 0} = \frac{-3 + 5}{1} = \frac{2}{1} = 2
\][/tex]
So, the slope [tex]\( m = 2 \)[/tex].
### Step 2: Determine the y-intercept ([tex]\( b \)[/tex])
1. The equation of a line in slope-intercept form is:
[tex]\[
y = mx + b
\][/tex]
2. To find the y-intercept ([tex]\( b \)[/tex]), we use one of the points and solve for [tex]\( b \)[/tex]. We'll use the point (0, -5):
[tex]\[
-5 = 2(0) + b
\][/tex]
[tex]\[
-5 = b
\][/tex]
So, the y-intercept [tex]\( b = -5 \)[/tex].
### Step 3: Write the function rule
Using the slope ([tex]\( m \)[/tex]) and y-intercept ([tex]\( b \)[/tex]), the equation of the line is:
[tex]\[
f(x) = 2x - 5
\][/tex]
Therefore, the function rule that describes the relationship between [tex]\( x \)[/tex] and [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = 2x - 5
\][/tex]
This equation represents the linear relationship shown in the table.
