Rotating a parabola involves applying a geometric transformation to every point on the curve, resulting in a new parabola oriented at a different angle. The fundamental principle is that if all points of a graph are rotated by the same angle, the entire graph rotates accordingly.
Understanding Parabola Rotation
Just like any other geometric shape, a parabola can be rotated around a specific point (typically the origin) by a chosen angle. This process transforms the coordinates of each point (x, y) on the original parabola to new coordinates (x', y') on the rotated parabola. By performing this transformation on the equation itself, we can derive the equation of the rotated parabola.
The Rotation Transformation Formulas
To rotate a point (x, y) about the origin by an angle θ (measured counter-clockwise), the new coordinates (x', y') are given by:
x' = x cos θ - y sin θy' = x sin θ + y cos θ
However, to rotate the equation of a parabola, say f(x, y) = 0, we need to substitute expressions for x and y from the original coordinate system in terms of x' and y' from the rotated coordinate system. This requires an inverse transformation:
| Original Coordinates (x, y) in terms of Rotated Coordinates (x', y') |
|---|
x = x' cos θ + y' sin θ |
y = -x' sin θ + y' cos θ |
These are the expressions you will substitute into the original parabola's equation.
Step-by-Step Guide to Rotating a Parabola's Equation
To rotate the equation of a parabola F(x, y) = 0 by an angle θ around the origin:
- Identify the Original Equation: Start with the standard equation of your parabola. For example, a common upright parabola is
y = ax^2 + bx + corx = ay^2 + by + c. For simplicity, let's consider a parabola with its vertex at the origin, such asy = x^2. - Choose the Angle of Rotation (θ): Determine the angle by which you want to rotate the parabola. Remember that a positive
θtypically indicates a counter-clockwise rotation. - Substitute the Transformation Formulas: In your parabola's equation, replace every instance of
xwith(x cos θ + y sin θ)and every instance ofywith(-x sin θ + y cos θ). - Simplify the Resulting Equation: Expand and simplify the new equation. This will likely result in a more complex, general conic section equation of the form
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. The presence of thexyterm indicates a rotation.
Example: Rotating y = x^2 by 45 degrees
Let's rotate the parabola y = x^2 by an angle θ = 45° (or π/4 radians).
First, find the values for cos θ and sin θ:
cos 45° = √2 / 2sin 45° = √2 / 2
Now, apply the substitution formulas:
- Replace
xwith(x (√2 / 2) + y (√2 / 2)) = (√2 / 2)(x + y) - Replace
ywith(-x (√2 / 2) + y (√2 / 2)) = (√2 / 2)(-x + y)
Substitute these into the original equation y = x^2:
(√2 / 2)(-x + y) = ((√2 / 2)(x + y))^2
Simplify the equation:
(√2 / 2)(-x + y) = (2 / 4)(x + y)^2
(√2 / 2)(-x + y) = (1 / 2)(x^2 + 2xy + y^2)
Multiply both sides by 2 to clear the fraction:
√2 (-x + y) = x^2 + 2xy + y^2
Rearrange the terms to get the general form of the conic section equation:
x^2 + 2xy + y^2 + √2 x - √2 y = 0
This is the equation of the parabola y = x^2 rotated by 45 degrees counter-clockwise.
Key Considerations
- Center of Rotation: The method described assumes rotation around the origin
(0,0). If you need to rotate around a different point(h,k), you would first translate the parabola so(h,k)moves to the origin, perform the rotation, and then translate it back. - Angle Direction: A positive angle
θdenotes a counter-clockwise rotation, while a negative angle denotes a clockwise rotation. - Resulting Equation: The rotated parabola's equation will generally be a second-degree equation with an
xyterm, indicating its rotated orientation. This is the general form of a conic section.
For further exploration of conic sections and transformations, resources like Khan Academy and Wolfram MathWorld offer in-depth explanations.
