Math Problem Statement
Solution
The graph shown represents a transformation of the basic quadratic function .
Analyzing the Transformation:
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Reflection: The graph opens downward, indicating a reflection over the x-axis. This means the leading coefficient is negative.
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Vertical Stretch/Compression: The graph appears to be stretched vertically since it is narrower than the graph of .
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Horizontal Translation: The vertex of the graph is located at , indicating that the graph has been translated 2 units to the right.
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Vertical Translation: The vertex also indicates a translation 4 units up.
Formulating the Equation:
Given the transformations, the equation can be written as: where is the vertex of the parabola.
Substituting the values and , and considering the reflection over the x-axis (which makes negative), the equation becomes:
To determine the value of , observe that when , :
Thus, the equation of the transformed function is:
Would you like more details on any part of the process? Here are five related questions you might find useful:
- What are the effects of different values of on the shape of a parabola?
- How does the vertex form of a quadratic function help in identifying transformations?
- What are other possible transformations of a quadratic function?
- Can this method be applied to other types of functions, like cubic or exponential?
- How would the graph change if the quadratic function had a different vertex?
Tip: When identifying transformations, always start by locating the vertex and determining if the parabola opens upwards or downwards!
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Math Problem Analysis
Mathematical Concepts
Transformation of Functions
Quadratic Functions
Vertex Form of a Parabola
Formulas
g(x) = a(x-h)^2 + k
Quadratic Formula
Theorems
Vertex form of quadratic functions
Suitable Grade Level
Grades 9-12
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