Compute the slope of the secant line containing the points (1, f (1)) and D) on the graph of f (x) = (x – 5)(x – 2)(x +1)(x + 4). Show the calculatie
Compute the slope of the secant line containing the points (1, f (1)) and D) on the graph of f (x) = (x – 5)(x – 2)(x +1)(x + 4). Show the calculatie
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Task: Calculate the Slope of a Secant Line**
**Objective:** Compute the slope of the secant line containing the points \((1, f(1))\) and \((3, f(3))\) on the graph of the function \(f(x) = (x - 5)(x - 2)(x + 1)(x + 4)\). Show the calculation.
1. **Identify the Points:**
- First, find the value of the function \(f(x)\) at points \(x = 1\) and \(x = 3\) to determine the coordinates of the points on the graph.
2. **Calculate \(f(1)\):**
\[
f(1) = (1 - 5)(1 - 2)(1 + 1)(1 + 4)
\]
\[
f(1) = (-4)(-1)(2)(5)
\]
\[
f(1) = 40
\]
3. **Calculate \(f(3)\):**
\[
f(3) = (3 - 5)(3 - 2)(3 + 1)(3 + 4)
\]
\[
f(3) = (-2)(1)(4)(7)
\]
\[
f(3) = -56
\]
4. **Determine the Points:**
- The coordinates are \((1, 40)\) and \((3, -56)\).
5. **Compute the Slope of the Secant Line:**
- Use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
\[
m = \frac{-56 - 40}{3 - 1}
\]
\[
m = \frac{-96}{2}
\]
\[
m = -48
\]
**Conclusion:**
The slope of the secant line passing through the points \((1, f(1))\) and \((3, f(3))\) is \(-48\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F236de4c3-f237-402b-b0e0-efa1a8147b53%2F84abd9e4-fb26-4a9e-928c-fb7b9d9d22ab%2Ft0an5l9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Task: Calculate the Slope of a Secant Line**
**Objective:** Compute the slope of the secant line containing the points \((1, f(1))\) and \((3, f(3))\) on the graph of the function \(f(x) = (x - 5)(x - 2)(x + 1)(x + 4)\). Show the calculation.
1. **Identify the Points:**
- First, find the value of the function \(f(x)\) at points \(x = 1\) and \(x = 3\) to determine the coordinates of the points on the graph.
2. **Calculate \(f(1)\):**
\[
f(1) = (1 - 5)(1 - 2)(1 + 1)(1 + 4)
\]
\[
f(1) = (-4)(-1)(2)(5)
\]
\[
f(1) = 40
\]
3. **Calculate \(f(3)\):**
\[
f(3) = (3 - 5)(3 - 2)(3 + 1)(3 + 4)
\]
\[
f(3) = (-2)(1)(4)(7)
\]
\[
f(3) = -56
\]
4. **Determine the Points:**
- The coordinates are \((1, 40)\) and \((3, -56)\).
5. **Compute the Slope of the Secant Line:**
- Use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
\[
m = \frac{-56 - 40}{3 - 1}
\]
\[
m = \frac{-96}{2}
\]
\[
m = -48
\]
**Conclusion:**
The slope of the secant line passing through the points \((1, f(1))\) and \((3, f(3))\) is \(-48\).
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