To determine the function that represents [tex]\( f(x) \)[/tex] based on the given points, we follow these steps:
### Step 1: Identify Two Points
We are given the following points from the table:
- [tex]\((-5, -126)\)[/tex]
- [tex]\((4, 0)\)[/tex]
- [tex]\((9, 70)\)[/tex]
- [tex]\((16, 168)\)[/tex]
### Step 2: Calculate the Slope (m)
Select two points, let’s use [tex]\((-5, -126)\)[/tex] and [tex]\((4, 0)\)[/tex].
The formula for the slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the given points:
[tex]\[ m = \frac{0 - (-126)}{4 - (-5)} = \frac{126}{9} = 14 \][/tex]
So, the slope [tex]\( m \)[/tex] is 14.
### Step 3: Find the y-intercept (b)
We use the slope [tex]\( m \)[/tex] and one of the points to find the y-intercept [tex]\( b \)[/tex]. Let’s use the point [tex]\((4, 0)\)[/tex].
The equation of a line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
Substitute [tex]\( m = 14 \)[/tex], [tex]\( x = 4 \)[/tex], and [tex]\( y = 0 \)[/tex]:
[tex]\[ 0 = 14 \cdot 4 + b \][/tex]
[tex]\[ 0 = 56 + b \][/tex]
[tex]\[ b = -56 \][/tex]
So, the y-intercept [tex]\( b \)[/tex] is -56.
### Step 4: Write the Equation
Now, we have the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex]. Substitute these values into the slope-intercept form of a linear equation:
[tex]\[ f(x) = 14x - 56 \][/tex]
### Final Answer
Therefore, the function that represents [tex]\( f \)[/tex] is:
[tex]\[ f(x) = 14x - 56 \][/tex]
### Answer Selection
From the options provided in your question, the correct answer is:
[tex]\[ f(x) = 14x - 56 \][/tex]
So the complete function is:
[tex]\[ f(x) = 14 \text{ (京) } \][/tex]
Choose [tex]\( 14x - 56 \)[/tex].
