To find the linear function [tex]\( f(x) \)[/tex] that passes through the points [tex]\( (-4, 7) \)[/tex] and [tex]\( (-8, -2) \)[/tex], we need to determine the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex].
### Step 1: Calculate the Slope [tex]\( m \)[/tex]
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the coordinates of the given points [tex]\((x_1, y_1) = (-4, 7)\)[/tex] and [tex]\((x_2, y_2) = (-8, -2)\)[/tex]:
[tex]\[
m = \frac{-2 - 7}{-8 - (-4)}
\][/tex]
[tex]\[
m = \frac{-9}{-4}
\][/tex]
[tex]\[
m = 2.25
\][/tex]
### Step 2: Calculate the Y-Intercept [tex]\( b \)[/tex]
The equation of a line in slope-intercept form is:
[tex]\[
y = mx + b
\][/tex]
We can rearrange this equation to solve for [tex]\( b \)[/tex]:
[tex]\[
b = y - mx
\][/tex]
Using the point [tex]\((-4, 7)\)[/tex] and the slope [tex]\( m = 2.25 \)[/tex]:
[tex]\[
b = 7 - (2.25 \times -4)
\][/tex]
[tex]\[
b = 7 + 9
\][/tex]
[tex]\[
b = 16
\][/tex]
### Step 3: Write the Equation of the Line
Now that we have the slope [tex]\( m = 2.25 \)[/tex] and the y-intercept [tex]\( b = 16 \)[/tex], we can write the linear function as:
[tex]\[
f(x) = 2.25x + 16
\][/tex]
So, the linear function that passes through the points [tex]\( (-4, 7) \)[/tex] and [tex]\( (-8, -2) \)[/tex] is:
[tex]\[
f(x) = 2.25x + 16
\][/tex]
