Which set of transformations is needed to graph $f\left(x\right)=-2\sin \left(x\right)+3$ from the parent sine function?

• A. Vertical compression by $2$, reflection over the y-axis, and vertical shift down $3$ units
• B. Vertical stretch by $2$, reflection over the y-axis, and vertical shift up $3$ units
• C. Horizontal stretch by $2$, reflection over the x-axis, and vertical shift up $3$ units
• D. Vertical stretch by $2$, reflection over the x-axis, and vertical shift up $3$ units

Answer: (D)

Verified step by step guidance

1. Identify the parent function, which is \(\sin(x)\).
2. Look at the coefficient in front of the sine function, which is \(-2\). The absolute value \$2$ indicates a vertical stretch by a factor of 2 because it multiplies the output values of \(\sin(x)\).
3. The negative sign in front of the 2 indicates a reflection over the x-axis, as it changes the sign of the output values of the sine function.
4. The constant term \(+3\) outside the sine function represents a vertical shift upward by 3 units, moving the entire graph up.
5. Combine these transformations: vertical stretch by 2, reflection over the x-axis, and vertical shift up by 3 units to obtain the graph of \(f(x) = -2\sin(x) + 3\) from the parent sine function.