(ROTASI FUNGSI) TRANSFORMASI MATEMATIKA SMA KELAS 11 #matematikasma #bukupaket #transformasigeometri

Summary

This tutorial explains how to rotate functions geometrically using a matrix-based approach. The process involves applying rotation transformations on both points and functions, adjusting for different angles (e.g., 90°, 180°, 270°), and rotating about various centers. Detailed steps show how to transform linear equations and parabolas through rotation, with an emphasis on handling the associated trigonometric identities and matrix operations. By following these steps, users will learn how to find the new coordinates of a rotated function and apply the transformation formula to solve related problems.

Takeaways

• 😀 Rotating a function involves applying rotation matrices to the coordinates of a point or the function itself.
• 😀 A rotation of 90° clockwise results in a negative sign for the trigonometric functions used in the matrix, while counterclockwise results in a positive sign.
• 😀 When rotating around the origin (0,0), the rotation matrix simplifies, and the coordinates (x, y) are transformed into new coordinates (x', y') using the rotation formula.
• 😀 The formula for rotation involves cosine and sine functions, where for a 90° clockwise rotation, cos(90°) = 0 and sin(90°) = -1.
• 😀 When rotating a function, the expressions for x and y need to be replaced with their respective transformed coordinates, e.g., x' = y and y' = -x after a 90° clockwise rotation.
• 😀 For a given rotation, always remember that the center of rotation (P, Q) affects the shift in coordinates before applying the rotation matrix.
• 😀 The center of rotation can be any point, and the corresponding rotation matrix needs to be adjusted by subtracting the coordinates of the center before applying the rotation formula.
• 😀 After applying the rotation, the function’s equation must be rewritten with transformed coordinates, and the rotation matrix needs to be used to simplify the equation.
• 😀 For example, after rotating the function 3y - 4x - 6 = 0 with a center of rotation at (1, 1), the new equation becomes 3x + 4y = 0 after simplifying.
• 😀 It is crucial to follow step-by-step procedures when rotating different types of functions (e.g., linear, quadratic) as each function's rotation might require a slightly different approach depending on the given parameters.

Q & A

What is the first step when rotating a function as described in the transcript?

The first step is to recall the rotation formula for a point, which involves using the rotation matrix. For a point with coordinates (x, y), the rotated coordinates (x', y') are determined by multiplying the original coordinates by the rotation matrix.

How does the sign of the angle affect the rotation direction in the script?

The script mentions that a negative angle results in a clockwise rotation, while a positive angle results in a counterclockwise rotation. This is important for determining the direction of the rotation, especially when dealing with functions.

What does the matrix for rotation look like in this context?

The rotation matrix is represented as: [cos(α) -sin(α)] [sin(α) cos(α)]. This matrix is used to apply the rotation to a set of coordinates.

What happens when you apply a 90° clockwise rotation to the equation 2y - 3x + 6 = 0?

When applying a 90° clockwise rotation to the equation 2y - 3x + 6 = 0, the result is a new equation: 2x + 3y + 6 = 0. The signs of the terms change according to the rotation rules and the transformation of the coordinates.

How do you rotate a function around a point that is not the origin?

To rotate a function around a point (p, q), you adjust the coordinates by subtracting the values of p and q from the original x and y coordinates before applying the rotation matrix. After the rotation, you add back the values of p and q to get the new coordinates.

What is the role of the center of rotation in the matrix calculations?

The center of rotation (p, q) plays a crucial role in adjusting the coordinates before applying the rotation matrix. The translation step ensures that the function or point rotates around the correct center by shifting the function or point accordingly.

What is the result when rotating the function 3y - 4x - 6 = 0 by 90° counterclockwise around the point (1, 1)?

When rotating the function 3y - 4x - 6 = 0 by 90° counterclockwise around the point (1, 1), the resulting equation is 3x + 4y = 0. This equation is derived by applying the rotation matrix and performing necessary translations.

How is the equation of a parabola affected by a 180° rotation?

A 180° rotation flips the parabola upside down. The transformed equation becomes y = -x² + 6x - 16, which is the result of applying the rotation formula and adjusting the terms accordingly.

What steps should be taken when rotating a function of the form y = x² - 2x + 6 by 180°?

First, the coordinates are translated based on the center of rotation. The rotation matrix is applied, and the new coordinates are calculated. Then, the values are substituted back into the function, and the expression is simplified to obtain the final rotated equation.

Why is it important to understand the negative angles in trigonometric calculations during rotation?

Understanding negative angles is important because they indicate a clockwise rotation. When using trigonometric functions such as sine and cosine, the sign of the angle affects the direction of the rotation. This understanding ensures that the rotation behaves as expected, especially for clockwise rotations.