At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Experience the ease of finding accurate answers to your questions from a knowledgeable community of professionals. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A 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 you found what you were looking for. Feel free to revisit us for more answers and updated information. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.