Answer :
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.
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.
