Math Problem Statement
Finish it
Solution
The text in the image is a math problem about a transformation of a parabola. Here's a translation of the problem into English:
"If a parabola with the equation is rotated around the origin (0, 0) by 90°, and then shifted downward by 3 units, what is the equation of the new parabola?"
Step-by-Step Solution:
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Initial Equation: The original equation is .
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Rotation by 90°:
- Rotating a point counterclockwise by 90° around the origin transforms the coordinates as follows: \quad \text{and} \quad y' = x$$
- Substituting these into the equation , we get: \quad \text{or} \quad x' = -x^2 + 2$$ and $$y' = x$$.
So the new equation after rotation is:
\quad \text{or} \quad x = -y^2 + 2$$ -
Shifting Down by 3 Units:
- A vertical shift by 3 units downward affects the -coordinate. If we translate the equation downward by 3 units, we subtract 3 from , yielding:
- So, the new equation becomes:
Thus, after a 90° rotation and shifting downward by 3 units, the new equation is:
Let me know if you'd like further clarification!
Related Questions:
- What would happen to the equation if the rotation was 180° instead of 90°?
- How would the equation change if the parabola was shifted upward by 3 units instead of downward?
- What are the general transformation rules for rotating conic sections?
- Can this transformation be done using matrix operations? How?
- What is the new vertex of the parabola after the transformation?
Tip:
Always consider transformations step-by-step (like rotation, translation) in the correct order to avoid confusion.
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Math Problem Analysis
Mathematical Concepts
Parabola
Coordinate Geometry
Transformations
Formulas
Rotation transformation: x' = -y, y' = x
Vertical translation: y = y' - k
Theorems
Rotation of Coordinates
Translation of Coordinates
Suitable Grade Level
Grades 10-12
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