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 \( y = \sin(x) \). For a function \( y = A\sin(x)+B \):

• The coefficient \( A \) affects vertical stretch/compression and reflection:
• If \( |A|>1 \), it's a vertical stretch by factor \( |A| \); if \( 0<|A|<1 \), vertical compression.
• If \( A<0 \), reflection across the \( x \)-axis.
• The constant \( B \) is vertical translation: \( B>0 \) up, \( B<0 \) down.

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

• For \( A = -2 \): \( |A| = 2>1 \), so vertical stretch by factor 2. The negative sign means reflection across the \( x \)-axis.
• For \( B = 3 \): vertical translation 3 units up.

Now check options:

• Option 1: Wrong (compression, reflection across \( y \)-axis).
• Option 2: Wrong (compression, down, \( x \)-reflection but other parts wrong).
• Option 3: Matches (reflection across \( x \)-axis, stretch by 2, translation up 3).
• Option 4: Wrong (reflection across \( y \)-axis, down).

Answer:

C. reflection across the x - axis, vertical stretching by a factor of 2, vertical translation 3 units up (assuming the third option is labeled C; if original labels are different, adjust, but based on content, the third option is correct)