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 the problem, we need to analyze the transformations applied to the function \( y = 4 \sin(x) \) and determine the new parameters after these transformations. ### Step 1: Identify the transformations 1. **Reflection in the \( x \)-axis**: This transformation changes the function from \( y = 4 \sin(x) \) to \( y = -4 \sin(x) \). 2. **Vertical stretch**: A vertical stretch by a factor of \( k \) (where \( k > 1 \)) will multiply the function by \( k \). If we assume the vertical stretch factor is \( k \), the function becomes \( y = -4k \sin(x) \). ### Step 2: Determine the values of \( a, b, c \) The transformed function can be expressed in the form \( y = a \sin(bx + c) \). From the transformations: - The amplitude \( a \) is \( -4k \). - The frequency \( b \) remains \( 1 \) since there is no horizontal stretch or compression. - The phase shift \( c \) is \( 0 \) since there is no horizontal shift. Thus, we have: - \( a = -4k \) - \( b = 1 \) - \( c = 0 \) ### Step 3: Sketch the graph To sketch the graph of the transformed function \( y = -4 \sin(x) \) (assuming \( k = 1 \) for simplicity), we note the following: - The amplitude is \( 4 \) (but reflected, so it goes downwards). - The period is \( 2\pi \). - The graph will oscillate between \( 0 \) and \( -4 \). ### Step 4: Calculate the exact \( x \)-coordinates of the \( x \)-axis intersections To find the \( x \)-coordinates where the graph intersects the \( x \)-axis, we set the function equal to zero: \[ -4 \sin(x) = 0 \] This simplifies to: \[ \sin(x) = 0 \] The solutions to this equation are: \[ x = n\pi \quad \text{for } n \in \mathbb{Z} \] ### Summary of Results a. The values are: - \( a = -4k \) (where \( k \) is the vertical stretch factor) - \( b = 1 \) - \( c = 0 \) b. The graph of the image is a sine wave oscillating between \( 0 \) and \( -4 \) with a period of \( 2\pi \). c. The exact \( x \)-coordinates of the intersections with the \( x \)-axis are \( x = n\pi \) for \( n \in \mathbb{Z} \).