### Transformation of Sine Functions The image displays a series of transformations applied to the sine function, \( \sin(x) \), along with arrows pointing to blank areas where the corresponding graphs would be illustrated. Here is a detailed transcription of the text and potential explanation that might appear on an educational website: #### Original Function \( f(x) = \sin(-x) \) - **Description:** This is the reflection of the original sine function across the y-axis. The sine function is an odd function, meaning \(\sin(-x) = -\sin(x)\). #### Transformation 1 \( f(x) = \sin\left(\frac{1}{2}x\right) \) - **Description:** This transforms the sine function by horizontally stretching it by a factor of 2. The period changes from \(2\pi\) to \(4\pi\). #### Transformation 2 \( f(x) = \sin(2x) \) - **Description:** This transforms the sine function by horizontally compressing it by a factor of 1/2. The period changes from \(2\pi\) to \(\pi\). ### Graphical Explanation: 1. **First Transformation Graph:** A graph demonstrating the reflection of the sine function across the y-axis. Since \(\sin(-x) = -\sin(x)\), the wave will be inverted around the y-axis. 2. **Second Transformation Graph:** A graph showing the sine function which has been stretched horizontally. This means there will be fewer cycles over the same x-interval, effectively doubling the period of the function. 3. **Third Transformation Graph:** A graph depicting the sine function which has been compressed horizontally. Here, the cycles of the sine wave are more frequent over the same x-interval, halving the period of the function. These transformations help to understand how various operations affect the shape and position of sine functions in a coordinate plane. ### Transformation of the Parent Sine Function Each function shown below is a transformation of the parent sine function. Based on the period, determine which graph represents each transformed function. #### Graph 1 (Top-Left): - The graph exhibits increased frequency, indicating that it has a shorter period compared to the parent sine function. - The graph completes more cycles within the same x-interval. #### Graph 2 (Top-Right): - This graph appears to have a period longer than that of the parent sine function. - The sine wave stretches horizontally, completing fewer cycles within the same x-interval. #### Graph 3 (Bottom-Left): - The sine wave in this graph closely resembles the parent sine function with no noticeable horizontal stretching or compressing. #### Graph 4 (Bottom-Right): - Similar to the graph in the top right, this graph also shows a stretched sine wave, indicating an increased period. #### Key Points: - The period of a sine function is determined by the frequency of its cycles. - A compressed sine wave indicates a transformation with a shorter period. - A stretched sine wave indicates a transformation with a longer period. - Horizontal translations and reflections are not the focus here; only the period is considered.