Certainly! Let's derive the equations for the linear functions based on the given data points.
### Question 4: Finding the linear function for [tex]\( f(0) = -2 \)[/tex] and [tex]\( f(8) = 4 \)[/tex]
We are given two points: [tex]\((0, -2)\)[/tex] and [tex]\((8, 4)\)[/tex].
1. Find the slope (m):
The formula to find the slope between 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]
Using the points [tex]\((0, -2)\)[/tex] and [tex]\((8, 4)\)[/tex]:
[tex]\[
m = \frac{4 - (-2)}{8 - 0} = \frac{4 + 2}{8} = \frac{6}{8} = 0.75
\][/tex]
2. Find the y-intercept (b):
The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex]. We can use one of the points to find [tex]\( b \)[/tex].
Using the point [tex]\((0, -2)\)[/tex]:
[tex]\[
-2 = (0.75 \cdot 0) + b \implies b = -2
\][/tex]
3. Write the equation:
Substituting [tex]\( m = 0.75 \)[/tex] and [tex]\( b = -2 \)[/tex] into the slope-intercept form, we get:
[tex]\[
f(x) = 0.75x - 2
\][/tex]
### Question 5: Finding the linear function for [tex]\( f(-3) = 6 \)[/tex] and [tex]\( f(0) = 5 \)[/tex]
We are given two points: [tex]\((-3, 6)\)[/tex] and [tex]\((0, 5)\)[/tex].
1. Find the slope (m):
The formula to find the slope between 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]
Using the points [tex]\((-3, 6)\)[/tex] and [tex]\((0, 5)\)[/tex]:
[tex]\[
m = \frac{5 - 6}{0 + 3} = \frac{-1}{3} = -0.333
\][/tex]
2. Find the y-intercept (b):
The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex]. We can use one of the points to find [tex]\( b \)[/tex].
Using the point [tex]\((0, 5)\)[/tex]:
[tex]\[
5 = (-0.333 \cdot 0) + b \implies b = 5
\][/tex]
3. Write the equation:
Substituting [tex]\( m = -0.333 \)[/tex] and [tex]\( b = 5 \)[/tex] into the slope-intercept form, we get:
[tex]\[
f(x) = -0.333x + 5
\][/tex]
### Summary of the Equations:
4. The equation for the linear function [tex]\( f \)[/tex] passing through [tex]\((0, -2)\)[/tex] and [tex]\((8, 4)\)[/tex] is:
[tex]\[
f(x) = 0.75x - 2
\][/tex]
5. The equation for the linear function [tex]\( f \)[/tex] passing through [tex]\((-3, 6)\)[/tex] and [tex]\((0, 5)\)[/tex] is:
[tex]\[
f(x) = -0.333x + 5
\][/tex]
