Math Problem Statement
Solution
Let's analyze the given graph, which is a transformation of the parent function . The parent function of a parabola, , has a vertex at and opens upwards. From the graph you provided, we can identify the following transformations:
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Vertical shift: The vertex of the parabola has moved to from the origin. This indicates a horizontal shift of 2 units to the left and a vertical shift of 3 units up.
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Reflection: The parabola opens downward, suggesting a reflection over the x-axis.
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Vertical stretch or compression: There seems to be no vertical stretch or compression based on the scale of the graph, so the vertical scaling factor remains unchanged.
Given these transformations, the general form for a transformed quadratic function is: where is the vertex, and controls the vertical stretch and the direction of the parabola.
- The vertex is , so and .
- Since the parabola is reflected over the x-axis, .
Thus, the equation for the function is:
Would you like further details on this or have any questions? Here are some related questions for deeper understanding:
- How does changing the value of affect the shape of a parabola?
- How would the equation change if the parabola were stretched vertically?
- What happens to the vertex of the parabola if you change the value of ?
- Can you describe what happens if only changes in the equation of a parabola?
- How does the horizontal shift impact the graph of a quadratic function?
Tip: A reflection over the x-axis occurs when the leading coefficient in the quadratic equation is negative.
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Math Problem Analysis
Mathematical Concepts
Algebra
Transformations of Functions
Quadratic Functions
Formulas
General form of a quadratic function: f(x) = a(x - h)^2 + k
Transformations of parabolas: horizontal shift, vertical shift, reflection
Theorems
Quadratic Function Vertex Form
Suitable Grade Level
Grades 9-11