Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![The graph above is a transformation of the function \( x^2 \).
Write an equation for the function graphed above:
\[ g(x) = \quad \boxed{} \]
### Graph Explanation
The graph shown is a parabola that opens downwards. It is centered on the point \((-1, 2)\). This suggests that the graph is a vertical and horizontal shift of the basic \( x^2 \) function, combined with a reflection across the x-axis.
1. **Vertex**: The vertex of the parabola is at \((-1, 2)\).
2. **Opening Direction**: The parabola opens downward, indicating a negative leading coefficient.
### Form of the Transformed Function
Given the transformations:
\[ g(x) = a(x-h)^2 + k \]
Where \( (h, k) \) is the vertex.
For this graph:
- \( h = -1 \)
- \( k = 2 \)
- Since the parabola opens downward, \( a \) will be negative.
Thus, a general form of the function could be written as:
\[ g(x) = -a(x + 1)^2 + 2 \]
Properly identifying the value of \( a \) could be determined if more points on the graph were provided, which typically describe the steepness or stretch/compression of the parabola.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88d18a19-21f4-41fb-afe0-306734676977%2F89e9a1a0-79c8-4fbf-bac3-9c21b65259e6%2Fk7la3xa.png&w=3840&q=75)
Transcribed Image Text:The graph above is a transformation of the function \( x^2 \).
Write an equation for the function graphed above:
\[ g(x) = \quad \boxed{} \]
### Graph Explanation
The graph shown is a parabola that opens downwards. It is centered on the point \((-1, 2)\). This suggests that the graph is a vertical and horizontal shift of the basic \( x^2 \) function, combined with a reflection across the x-axis.
1. **Vertex**: The vertex of the parabola is at \((-1, 2)\).
2. **Opening Direction**: The parabola opens downward, indicating a negative leading coefficient.
### Form of the Transformed Function
Given the transformations:
\[ g(x) = a(x-h)^2 + k \]
Where \( (h, k) \) is the vertex.
For this graph:
- \( h = -1 \)
- \( k = 2 \)
- Since the parabola opens downward, \( a \) will be negative.
Thus, a general form of the function could be written as:
\[ g(x) = -a(x + 1)^2 + 2 \]
Properly identifying the value of \( a \) could be determined if more points on the graph were provided, which typically describe the steepness or stretch/compression of the parabola.
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