which set of transformations is needed to graph f(x) = -2sin(x) + 3 from the parent sine function?\n○…

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

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

Answer

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).