Chapter3: Functions
Section3.5: Transformation Of Functions
Problem 5SE: How can you determine whether a function is odd or even from the formula of the function?
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![**Tangent Line to a Function**
*Problem Statement:*
Find the equation of the line tangent to the graph of \( f(x) = 3 - 2x \) at \( x = 3 \).
*Solution:*
To find the equation of the tangent line to the function \( f(x) = 3 - 2x \) at \( x = 3 \), follow these steps:
1. **Calculate the derivative \( f'(x) \):**
The derivative \( f'(x) \) represents the slope of the tangent line to the function at any point \( x \). For the function \( f(x) = 3 - 2x \):
\( f'(x) = -2 \)
This indicates that the slope of the tangent line at any point on the graph of \( f(x) \) is -2.
2. **Evaluate \( f(x) \) at \( x = 3 \):**
To find the y-coordinate of the point where the tangent line touches the graph, substitute \( x = 3 \) into the function \( f(x) \):
\( f(3) = 3 - 2(3) = 3 - 6 = -3 \)
So, the point on the graph where the tangent line touches is (3, -3).
3. **Use the point-slope form of the equation of a line:**
The point-slope form of the equation of a line is given by:
\[ y - y_1 = m(x - x_1) \]
where \( m \) is the slope and \( (x_1, y_1) \) is the point on the line.
Given \( m = -2 \) and the point (3, -3):
\[ y - (-3) = -2(x - 3) \]
Simplify the equation:
\[ y + 3 = -2(x - 3) \]
\[ y + 3 = -2x + 6 \]
\[ y = -2x + 6 - 3 \]
\[ y = -2x + 3 \]
Therefore, the equation of the tangent line is:
\[ y = -2x + 3 \]
*Answer:*
\( y = -2x + 3 \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5ea397a-632d-4020-8701-e2de67ac8a17%2Fd4dccdf3-646f-4c71-ba43-ae29a1c16531%2Ftrou9u4_processed.png&w=3840&q=75)
Transcribed Image Text:**Tangent Line to a Function**
*Problem Statement:*
Find the equation of the line tangent to the graph of \( f(x) = 3 - 2x \) at \( x = 3 \).
*Solution:*
To find the equation of the tangent line to the function \( f(x) = 3 - 2x \) at \( x = 3 \), follow these steps:
1. **Calculate the derivative \( f'(x) \):**
The derivative \( f'(x) \) represents the slope of the tangent line to the function at any point \( x \). For the function \( f(x) = 3 - 2x \):
\( f'(x) = -2 \)
This indicates that the slope of the tangent line at any point on the graph of \( f(x) \) is -2.
2. **Evaluate \( f(x) \) at \( x = 3 \):**
To find the y-coordinate of the point where the tangent line touches the graph, substitute \( x = 3 \) into the function \( f(x) \):
\( f(3) = 3 - 2(3) = 3 - 6 = -3 \)
So, the point on the graph where the tangent line touches is (3, -3).
3. **Use the point-slope form of the equation of a line:**
The point-slope form of the equation of a line is given by:
\[ y - y_1 = m(x - x_1) \]
where \( m \) is the slope and \( (x_1, y_1) \) is the point on the line.
Given \( m = -2 \) and the point (3, -3):
\[ y - (-3) = -2(x - 3) \]
Simplify the equation:
\[ y + 3 = -2(x - 3) \]
\[ y + 3 = -2x + 6 \]
\[ y = -2x + 6 - 3 \]
\[ y = -2x + 3 \]
Therefore, the equation of the tangent line is:
\[ y = -2x + 3 \]
*Answer:*
\( y = -2x + 3 \)
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