Example 8 The graph of \( y=4 \sin (x) \) undergoes a dita reflection in the \( x \)-axis, followed by a verti After the three transformations are applie a Determine the values of \( a, b \) and \( c \). b Sketch the graph of the image for \( x \in \) c Calculate the exact \( x \)-coordinates of \( x \)-axis.

To solve this, we’ll perform the transformations step by step. 1. **Dita Reflection in the \( x \)-axis**: This will change the equation to \( y = -4 \sin(x) \). 2. **Vertical Stretch**: If we assume this is a stretch by a factor \( k \), say \( k=2 \), the equation becomes \( y = -8 \sin(x) \). Thus, \( a = -8 \), \( b = 1 \) (no horizontal scaling), and \( c = 0 \) (no horizontal shifting). Now, the values are: - \( a = -8 \) - \( b = 1 \) - \( c = 0 \) **Sketch of the graph**: The graph of \( y = -8 \sin(x) \) would follow the sinusoidal pattern flipped upside down. It has amplitude of 8 instead of 4, and will oscillate between -8 and 0, with the typical period of \( 2\pi \). **Finding exact \( x \)-coordinates of intersection with the \( x \)-axis**: To find where the graph intersects the \( x \)-axis, set \( y=0 \): \[ -8 \sin(x) = 0 \implies \sin(x) = 0 \] The solution to this is \( x = n\pi \) where \( n \) is any integer. Thus, the exact \( x \)-coordinates of intersection are \( 0, \pi, 2\pi, \ldots \)