Find an equation of the line tangent to the graph of f(x) = (5x − 9)(x + 6) at (2,8). The equation of the line tangent to the graph of f(x) = (5x-9)(x + 6) at (2,8) is (Type an equation.)
Find an equation of the line tangent to the graph of f(x) = (5x − 9)(x + 6) at (2,8). The equation of the line tangent to the graph of f(x) = (5x-9)(x + 6) at (2,8) is (Type an equation.)
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter3: Linear And Nonlinear Functions
Section: Chapter Questions
Problem 26MCQ
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![**Finding the Equation of a Tangent Line**
**Problem Statement:**
Find an equation of the line tangent to the graph of \( f(x) = (5x - 9)(x + 6) \) at \( (2,8) \).
---
**Solution:**
The equation of the line tangent to the graph of \( f(x) = (5x - 9)(x + 6) \) at \( (2,8) \) is \(\_\_\_\_\_\_\_\_\_\_\_ \).
(Type an equation.)
---
**Detailed Explanation:**
To find the equation of the tangent line, we need to follow the steps outlined below:
1. **Find the Derivative of \( f(x) \):**
- First, expand the function \( f(x) \):
\[
f(x) = (5x - 9)(x + 6) = 5x^2 + 30x - 9x - 54 = 5x^2 + 21x - 54
\]
- Now, take the derivative of \( f(x) \):
\[
f'(x) = \frac{d}{dx}(5x^2 + 21x - 54) = 10x + 21
\]
2. **Evaluate the Derivative at \( x = 2 \):**
\[
f'(2) = 10(2) + 21 = 20 + 21 = 41
\]
This gives us the slope of the tangent line.
3. **Use the Point-Slope Form of the Line:**
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
Where \( (x_1, y_1) \) is the point of tangency \( (2, 8) \), and \( m \) is the slope found above.
Therefore:
\[
y - 8 = 41(x - 2)
\]
4. **Simplify the Equation:**
\[
y - 8 = 41x - 82
\]
\[
y = 41x - 74
\]
Thus, the equation of the tangent line to the graph of \( f](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97184fab-fce4-43b1-85fd-c47064c00bbe%2F06446f19-bc18-4d59-9afd-428ede2ffad0%2F6djne6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Equation of a Tangent Line**
**Problem Statement:**
Find an equation of the line tangent to the graph of \( f(x) = (5x - 9)(x + 6) \) at \( (2,8) \).
---
**Solution:**
The equation of the line tangent to the graph of \( f(x) = (5x - 9)(x + 6) \) at \( (2,8) \) is \(\_\_\_\_\_\_\_\_\_\_\_ \).
(Type an equation.)
---
**Detailed Explanation:**
To find the equation of the tangent line, we need to follow the steps outlined below:
1. **Find the Derivative of \( f(x) \):**
- First, expand the function \( f(x) \):
\[
f(x) = (5x - 9)(x + 6) = 5x^2 + 30x - 9x - 54 = 5x^2 + 21x - 54
\]
- Now, take the derivative of \( f(x) \):
\[
f'(x) = \frac{d}{dx}(5x^2 + 21x - 54) = 10x + 21
\]
2. **Evaluate the Derivative at \( x = 2 \):**
\[
f'(2) = 10(2) + 21 = 20 + 21 = 41
\]
This gives us the slope of the tangent line.
3. **Use the Point-Slope Form of the Line:**
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
Where \( (x_1, y_1) \) is the point of tangency \( (2, 8) \), and \( m \) is the slope found above.
Therefore:
\[
y - 8 = 41(x - 2)
\]
4. **Simplify the Equation:**
\[
y - 8 = 41x - 82
\]
\[
y = 41x - 74
\]
Thus, the equation of the tangent line to the graph of \( f
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