Transformations Q1 \( \begin{array}{l}\text { Check the Plot Function box to graph } f(x)=\sin (x) \text {. Use the a-slider to graph the function } \\ f(x)=5 \sin (x) \text {. The value of a affects the } \\ \text { Use the interactive figure to find your answer. Use the left and right arrow keys to move } \\ \text { along a slider as needed. } \\ \text { Click here to launch the interactive figure. }\end{array} \) The value of a affects the 1

The value of \( a \) in the function \( f(x) = a \sin(x) \) affects the amplitude of the sine wave. 1. **Amplitude**: The amplitude is the maximum distance from the midline (y = 0) to the peak (or trough) of the wave. In this case, when \( a = 5 \), the amplitude of the function \( f(x) = 5 \sin(x) \) is 5. This means the graph will reach a maximum value of 5 and a minimum value of -5. 2. **Graph Behavior**: As you adjust the value of \( a \) using the slider, you will notice that: - If \( a > 1 \), the peaks and troughs of the sine wave will be higher and lower, respectively. - If \( a < 1 \), the peaks and troughs will be closer to the midline, resulting in a smaller amplitude. - If \( a = 0 \), the function will collapse to the x-axis, as \( f(x) = 0 \) for all \( x \). 3. **Period and Frequency**: The value of \( a \) does not affect the period or frequency of the sine function, which remains \( 2\pi \) for \( \sin(x) \). In summary, the value of \( a \) primarily affects the amplitude of the sine function, changing how "tall" or "short" the wave appears on the graph.