To determine the equation of the linear function [tex]\( f(x) \)[/tex] that fits the given points [tex]\( f(0) = 8 \)[/tex] and [tex]\( f(1) = 12 \)[/tex], we can follow these steps:
1. Identify the form of the linear function: A linear function can generally be written in the form [tex]\( f(x) = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
2. Calculate the slope [tex]\( m \)[/tex]:
The slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated as:
[tex]\[
m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
Using the points [tex]\((0, 8)\)[/tex] and [tex]\((1, 12)\)[/tex]:
[tex]\[
m = \frac{12 - 8}{1 - 0} = \frac{4}{1} = 4
\][/tex]
3. Determine the y-intercept [tex]\( b \)[/tex]:
The y-intercept [tex]\( b \)[/tex] is found by evaluating [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = b
\][/tex]
Given that [tex]\( f(0) = 8 \)[/tex], we have:
[tex]\[
b = 8
\][/tex]
4. Write the equation of the linear function:
Now, substituting the values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex] into the general form [tex]\( f(x) = mx + b \)[/tex], we get:
[tex]\[
f(x) = 4x + 8
\][/tex]
Finally, comparing this equation with the given options:
- A) [tex]\( f(x) = 12x + 8 \)[/tex] (Incorrect, slope is 12 which doesn't match).
- B) [tex]\( f(x) = 4x \)[/tex] (Incorrect, missing y-intercept b).
- C) [tex]\( f(x) = 4x + 12 \)[/tex] (Incorrect, y-intercept is wrong).
- D) [tex]\( f(x) = 4x + 8 \)[/tex] (Correct, matches our equation).
Thus, the correct answer is:
[tex]\[ \boxed{D} \ f(x) = 4x + 8 \][/tex]
