Question

which set of transformations is needed to graph f(x) = -2sin(x) + 3 from the parent sine function?
○ vertical compression by a factor of 2, vertical translation 3 units up, reflection across the y-axis
○ vertical compression by a factor of 2, vertical translation 3 units down, reflection across the x-axis
○ reflection across the x-axis, vertical stretching by a factor of 2, vertical translation 3 units up
○ reflection across the y-axis, vertical stretching by a factor of 2, vertical translation 3 units down

Explanation:

Step1: Recall Transformations of Sine Function

The parent sine function is \( y = \sin(x) \). The general form of a transformed sine function is \( y = A\sin(Bx - C) + D \), where:

• \( |A| \) is the vertical stretch/compression factor. If \( |A|>1 \), it's a stretch; if \( 0<|A|<1 \), it's a compression. A negative \( A \) indicates a reflection over the x - axis.
• \( D \) is the vertical translation (up if \( D>0 \), down if \( D<0 \)).

Step2: Analyze the Given Function \( f(x)=-2\sin(x)+3 \)

• For the amplitude and reflection: The coefficient of \( \sin(x) \) is - 2. The absolute value \( | - 2|=2 \), so there is a vertical stretch by a factor of 2. The negative sign indicates a reflection across the x - axis (since \( A=-2 \), and a negative \( A \) reflects over the x - axis).
• For the vertical translation: The constant term is 3, which means a vertical translation 3 units up (because \( D = 3>0 \)).

Now let's check the options:

• Option 1: Vertical compression by a factor of 2 (wrong, since \( |A| = 2>1 \), it's a stretch), vertical translation 3 units up, reflection across the y - axis (wrong, the reflection is across the x - axis).
• Option 2: Vertical compression by a factor of 2 (wrong), vertical translation 3 units down (wrong, \( D = 3>0 \) is up), reflection across the x - axis (partial correct).
• Option 3: Reflection across the x - axis (correct, because \( A=-2 \)), vertical stretching by a factor of 2 (correct, \( |A| = 2 \)), vertical translation 3 units up (correct, \( D = 3 \)).
• Option 4: Reflection across the y - axis (wrong), vertical stretching by a factor of 2 (correct), vertical translation 3 units down (wrong).

Answer:

The correct option is the one with "reflection across the x - axis, vertical stretching by a factor of 2, vertical translation 3 units up" (the third option in the list of options provided in the problem).