Which set of transformations is needed to graph [tex]f(x)=-2 \sin (x)+3[/tex] from the parent sine function?
A. vertical compression by a factor of 2, vertical translation 3 units up, reflection across the [tex]y[/tex]-axis
B. vertical compression by a factor of 2, vertical translation 3 units down, reflection across the [tex]x[/tex]-axis
C. reflection across the [tex]x[/tex]-axis, vertical stretching by a factor of 2, vertical translation 3 units up
D. reflection across the [tex]y[/tex]-axis, vertical stretching by a factor of 2, vertical translation 3 units down
Answer (2)
The function f ( x ) = − 2 sin ( x ) + 3 requires three transformations from the parent sine function: reflection across the x-axis, vertical stretching by a factor of 2, and vertical translation 3 units up. The answer is option C . These transformations alter the amplitude and position of the sine wave accordingly.
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The function f ( x ) = − 2 sin ( x ) + 3 is obtained from the parent function y = sin ( x ) by applying a series of transformations.
The coefficient -2 indicates a reflection across the x-axis and a vertical stretch by a factor of 2.
The constant term +3 indicates a vertical translation 3 units up.
Therefore, the required transformations are reflection across the x-axis, vertical stretching by a factor of 2, and vertical translation 3 units up. reflection across the x -axis, vertical stretching by a factor of 2, vertical translation 3 units up
Explanation
Understanding the Problem We are given the function f ( x ) = − 2 sin ( x ) + 3 and we want to describe the transformations needed to obtain its graph from the parent function y = sin ( x ) .
Identifying Transformations The function f ( x ) = − 2 sin ( x ) + 3 involves a vertical stretch, a reflection, and a vertical translation of the parent sine function. Let's break down each transformation.
Analyzing the Coefficient The coefficient − 2 in front of the sin ( x ) term indicates two transformations:
Vertical Stretch: A vertical stretch by a factor of 2.
Reflection across the x-axis: The negative sign indicates a reflection across the x-axis.
Analyzing the Constant Term The constant term + 3 indicates a vertical translation 3 units up.
Conclusion Therefore, the transformations needed to graph f ( x ) = − 2 sin ( x ) + 3 from the parent sine function y = sin ( x ) are:
Reflection across the x-axis.
Vertical stretching by a factor of 2.
Vertical translation 3 units up.
Examples
Understanding transformations of functions is crucial in many fields. For example, in physics, when studying wave phenomena, transformations like stretching or shifting can help model changes in amplitude or phase. In engineering, signal processing often involves transforming signals to analyze their frequency components. In computer graphics, transformations are used to manipulate and animate objects in 2D and 3D space. Knowing how to apply these transformations allows us to predict and control the behavior of various systems.
