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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![The problem states: "The graphs of \( f \) and \( g \) are given. Find a formula for the function \( g \)."
**Graph Explanation:**
- **Graph of \( f(x) = x^2 \):**
- The graph is a blue parabola opening upwards with its vertex at the origin (0, 0).
- The equation is \( f(x) = x^2 \), representing a standard quadratic function.
- **Graph of \( g(x) \):**
- The graph is a red parabola opening downwards.
- The vertex of this parabola is at approximately (3, 3), and it is symmetric about the vertical line \( x = 3 \).
- This implies a transformation of the standard parabola \( f(x) = x^2 \).
To find the formula for \( g(x) \), observe that:
- The vertex form of a parabola is \( g(x) = a(x-h)^2 + k \).
- Here, \( (h, k) = (3, 3) \).
- Since it opens downwards, \( a \) must be negative. By matching the points or using the graph, \( a \) can be determined.
Thus, the function for \( g(x) \) is given by:
\[ g(x) = -a(x-3)^2 + 3 \]
If you calculate the specific \( a \) value using points from the graph, you find that \( g(x) = -(x-3)^2 + 3 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25e588b9-b272-406a-8435-8ef7027d16be%2Fd1d8bb39-b5cc-4454-8144-6b23fc9c8aeb%2Fkynradb_processed.png&w=3840&q=75)
Transcribed Image Text:The problem states: "The graphs of \( f \) and \( g \) are given. Find a formula for the function \( g \)."
**Graph Explanation:**
- **Graph of \( f(x) = x^2 \):**
- The graph is a blue parabola opening upwards with its vertex at the origin (0, 0).
- The equation is \( f(x) = x^2 \), representing a standard quadratic function.
- **Graph of \( g(x) \):**
- The graph is a red parabola opening downwards.
- The vertex of this parabola is at approximately (3, 3), and it is symmetric about the vertical line \( x = 3 \).
- This implies a transformation of the standard parabola \( f(x) = x^2 \).
To find the formula for \( g(x) \), observe that:
- The vertex form of a parabola is \( g(x) = a(x-h)^2 + k \).
- Here, \( (h, k) = (3, 3) \).
- Since it opens downwards, \( a \) must be negative. By matching the points or using the graph, \( a \) can be determined.
Thus, the function for \( g(x) \) is given by:
\[ g(x) = -a(x-3)^2 + 3 \]
If you calculate the specific \( a \) value using points from the graph, you find that \( g(x) = -(x-3)^2 + 3 \).
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