Algebra: Structure And Method, Book 1
(REV)00 Edition Edition
ISBN: 9780395977224
Author: Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher: McDougal Littell
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![**Problem Statement:**
Find the linear function with the following properties.
\[ f(2) = 10 \]
\[ f(10) = -2 \]
---
**Answer:**
**2 Points**
\[ f(x) = \left|\begin{array}{c} \quad \end{array}\right. \]
---
*Note: In the given setup, the function \( f(x) \) is to be determined. Currently, a box is provided where the expression for the linear function should be entered.*
To solve for the linear function:
### Steps:
1. **Find the Slope (m):**
- Use the formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
- Here, \((x_1, y_1) = (2, 10)\) and \((x_2, y_2) = (10, -2)\).
- Calculate:
\[ m = \frac{-2 - 10}{10 - 2} = \frac{-12}{8} = -\frac{3}{2} \]
2. **Find the Equation Using Point-Slope Form:**
- Use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
- Substitute \( m = -\frac{3}{2} \), and one of the points, say \((2, 10)\):
\[ y - 10 = -\frac{3}{2}(x - 2) \]
3. **Simplify to Get the Function:**
- Simplify the equation to the slope-intercept form \( y = mx + b \):
\[ y - 10 = -\frac{3}{2}x + 3 \]
\[ y = -\frac{3}{2}x + 3 + 10 \]
\[ y = -\frac{3}{2}x + 13 \]
The linear function is:
\[ f(x) = -\frac{3}{2}x + 13 \](https://content.bartleby.com/qna-images/question/6a720606-2121-4432-823f-1c1dc1bab0c0/87c5fa9f-131b-4fea-9156-e7554cff99f5/bv1usee_processed.jpeg)
Transcribed Image Text:**Problem Statement:**
Find the linear function with the following properties.
\[ f(2) = 10 \]
\[ f(10) = -2 \]
---
**Answer:**
**2 Points**
\[ f(x) = \left|\begin{array}{c} \quad \end{array}\right. \]
---
*Note: In the given setup, the function \( f(x) \) is to be determined. Currently, a box is provided where the expression for the linear function should be entered.*
To solve for the linear function:
### Steps:
1. **Find the Slope (m):**
- Use the formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
- Here, \((x_1, y_1) = (2, 10)\) and \((x_2, y_2) = (10, -2)\).
- Calculate:
\[ m = \frac{-2 - 10}{10 - 2} = \frac{-12}{8} = -\frac{3}{2} \]
2. **Find the Equation Using Point-Slope Form:**
- Use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
- Substitute \( m = -\frac{3}{2} \), and one of the points, say \((2, 10)\):
\[ y - 10 = -\frac{3}{2}(x - 2) \]
3. **Simplify to Get the Function:**
- Simplify the equation to the slope-intercept form \( y = mx + b \):
\[ y - 10 = -\frac{3}{2}x + 3 \]
\[ y = -\frac{3}{2}x + 3 + 10 \]
\[ y = -\frac{3}{2}x + 13 \]
The linear function is:
\[ f(x) = -\frac{3}{2}x + 13 \
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