Looking for reliable answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Connect with professionals ready to provide precise answers to your questions on our comprehensive Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
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]
### 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]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.