Which equation represents a rotation of the equation $y=4 x+3$?
A. $y=3 x+2$
B. $y=-3 x-3$
C. $y=4 x+3$
D. $y=3 x+3$
Answer (2)
The problem provides a line y = 4 x + 3 and asks to identify a rotation of this line from the given options.
A rotation of a line generally changes its slope, except for a 180-degree rotation which keeps the slope the same but changes the y-intercept.
Examine each option and compare its slope to the original line's slope (4). Any option with a different slope represents a rotation.
The equation y = 3 x + 2 represents a rotation of the original line. y = 3 x + 2
Explanation
Analyze the problem The given equation is y = 4 x + 3 . We need to identify which of the given options represents a rotation of this line. A rotation of a line generally changes its slope. The only exception is a rotation by 180 degrees, which results in a line with the same slope but a different y-intercept, or the same line if the rotation is around a point on the line.
Examine the options Let's examine the given options:
Option 1: y = 3 x + 2 . The slope is 3, which is different from 4. This represents a rotation.
Option 2: y = − 3 x − 3 . The slope is -3, which is different from 4. This represents a rotation.
Option 3: y = 4 x + 3 . The slope is 4 and the y-intercept is 3, which is the same as the original line. This is the same line, not a rotation.
Option 4: y = 3 x + 3 . The slope is 3, which is different from 4. This represents a rotation.
Identify the rotation Since the question asks for which equation represents a rotation, any equation with a different slope represents a rotation. Therefore, y = 3 x + 2 , y = − 3 x − 3 and y = 3 x + 3 are all rotations of the original line. However, since we are asked to choose only one equation, we can assume that the rotation is not by 180 degrees. In this case, the slope must be different. Let's choose the first option.
Examples
Understanding rotations of lines is crucial in various fields, such as computer graphics, where objects are rotated on a screen. For instance, when designing a video game, developers use rotations to change the orientation of characters or objects, making the game more dynamic and visually appealing. By applying mathematical principles of linear transformations, they ensure that these rotations are performed accurately and efficiently, enhancing the overall gaming experience.
The equation that represents a rotation of the original line y = 4 x + 3 is y = 3 x + 2 because it has a different slope. There are also other valid rotations, but Option A serves as a clear example. Thus, the chosen option is A.
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