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
![**Function Transformations**
Given the following function:
\[ f(x) = -\frac{x^2}{4} + 4 \]
Determine the more basic function that has been shifted, reflected, stretched, or compressed.
**Analysis and Explanation:**
The basic function here is \( f(x) = x^2 \), which is a standard parabola opening upwards with its vertex at the origin (0, 0).
Transformations applied to this basic function to arrive at \( f(x) \) are as follows:
1. **Reflection:**
- The negative sign in front of the fraction \( -\frac{x^2}{4} \) indicates that the function is reflected over the x-axis. Therefore, instead of opening upwards as \( x^2 \) does, the parabola opens downwards.
2. **Vertical Compression:**
- The term \(\frac{x^2}{4}\) suggests a vertical compression. In the original function \( x^2 \), the coefficient of \( x^2 \) is 1. Here it is \(\frac{1}{4}\), meaning the parabola is compressed by a factor of 4.
3. **Vertical Shift:**
- The \( + 4 \) at the end of the function indicates a vertical shift upwards by 4 units. This moves the vertex of the parabola from (0, 0) to (0, 4).
Therefore, the function \( f(x) = -\frac{x^2}{4} + 4 \) is derived from the basic function \( x^2 \) by reflecting it over the x-axis, compressing it vertically by a factor of 4, and shifting it upwards by 4 units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4524aa64-b0b4-4d9f-9205-0e5a55516a84%2Fa451a4ff-5d0c-4455-b55e-51c7a4dfbcad%2Ftyxscmp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Function Transformations**
Given the following function:
\[ f(x) = -\frac{x^2}{4} + 4 \]
Determine the more basic function that has been shifted, reflected, stretched, or compressed.
**Analysis and Explanation:**
The basic function here is \( f(x) = x^2 \), which is a standard parabola opening upwards with its vertex at the origin (0, 0).
Transformations applied to this basic function to arrive at \( f(x) \) are as follows:
1. **Reflection:**
- The negative sign in front of the fraction \( -\frac{x^2}{4} \) indicates that the function is reflected over the x-axis. Therefore, instead of opening upwards as \( x^2 \) does, the parabola opens downwards.
2. **Vertical Compression:**
- The term \(\frac{x^2}{4}\) suggests a vertical compression. In the original function \( x^2 \), the coefficient of \( x^2 \) is 1. Here it is \(\frac{1}{4}\), meaning the parabola is compressed by a factor of 4.
3. **Vertical Shift:**
- The \( + 4 \) at the end of the function indicates a vertical shift upwards by 4 units. This moves the vertex of the parabola from (0, 0) to (0, 4).
Therefore, the function \( f(x) = -\frac{x^2}{4} + 4 \) is derived from the basic function \( x^2 \) by reflecting it over the x-axis, compressing it vertically by a factor of 4, and shifting it upwards by 4 units.
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