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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![### Example 5
**Objective:** Find an equation of the tangent line to the parabola \( y = x^2 - 8x + 9 \) at the point \( (1, 2) \).
**Solution:** From the previous example, we know the derivative of \( f(x) = x^2 - 8x + 9 \) at the number \( a \) is \( f'(a) = 2a - 8 \). Therefore the slope of the tangent line at \( (1, 2) \) is \( f'(1) = 2(1) - 8 = -6 \).
Thus an equation of the tangent line, as shown in the figure, is
\[
y - (2) = -6 (x - 1)
\]
or
\[
y = -6x + 6 + 2
\]
or
\[
y = -6x + 8
\]
**Explanation of the Graph:**
The graph on the left visually represents a parabola and its tangent line. The horizontal axis is labeled as \( x \) and the vertical axis is labeled as \( y \). The parabola \( y = x^2 - 8x + 9 \) is drawn with its vertex and a portion of its arms visible, colored in blue. The point \( (1, 2) \) on the parabola is marked, and the tangent line at this point is drawn in red. The tangent line intersects the parabola at \( (1, 2) \) and represents the linear equation we derived: \( y = -6x + 8 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb9de090-ec28-43d0-bfa1-94c200024acb%2F720d3ef7-d2fe-4485-aea2-f8cf0c3f0d91%2Fldq8adr_processed.png&w=3840&q=75)
Transcribed Image Text:### Example 5
**Objective:** Find an equation of the tangent line to the parabola \( y = x^2 - 8x + 9 \) at the point \( (1, 2) \).
**Solution:** From the previous example, we know the derivative of \( f(x) = x^2 - 8x + 9 \) at the number \( a \) is \( f'(a) = 2a - 8 \). Therefore the slope of the tangent line at \( (1, 2) \) is \( f'(1) = 2(1) - 8 = -6 \).
Thus an equation of the tangent line, as shown in the figure, is
\[
y - (2) = -6 (x - 1)
\]
or
\[
y = -6x + 6 + 2
\]
or
\[
y = -6x + 8
\]
**Explanation of the Graph:**
The graph on the left visually represents a parabola and its tangent line. The horizontal axis is labeled as \( x \) and the vertical axis is labeled as \( y \). The parabola \( y = x^2 - 8x + 9 \) is drawn with its vertex and a portion of its arms visible, colored in blue. The point \( (1, 2) \) on the parabola is marked, and the tangent line at this point is drawn in red. The tangent line intersects the parabola at \( (1, 2) \) and represents the linear equation we derived: \( y = -6x + 8 \).
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