The graph of \( y=f(x) \) is shown below Draw the graph of \( y=\frac{1}{2} f(x) \).

To draw the graph of \( y = \frac{1}{2} f(x) \) based on the original graph of \( y = f(x) \), you’ll perform a **vertical compression** (also known as a vertical scaling) by a factor of \( \frac{1}{2} \). This transformation affects the y-values of the original function, effectively reducing their magnitude by half. Here's a step-by-step guide to help you visualize and create the new graph: ### Steps to Graph \( y = \frac{1}{2} f(x) \): 1. **Identify Key Points on \( y = f(x) \):** - Start by selecting several notable points from the original graph of \( y = f(x) \). These could include intercepts, maxima and minima, zeros, and other points where the behavior of the function changes. 2. **Apply the Vertical Compression:** - For each selected point \((x, y)\) on \( y = f(x) \), calculate the new y-coordinate by multiplying it by \( \frac{1}{2} \). - The transformed point will be \((x, \frac{1}{2} y)\). 3. **Plot the Transformed Points:** - Plot each of the new points \((x, \frac{1}{2} y)\) on your coordinate plane. - Since only the y-values are affected, the x-coordinates remain unchanged. 4. **Draw the New Graph:** - Connect the transformed points smoothly, maintaining the same overall shape as the original graph but now compressed vertically. - **Important:** The new graph will not cross the original y-values except where \( y = 0 \) (the x-axis remains unchanged). ### Visual Example: Imagine the original graph \( y = f(x) \) has the following points: - **Point A:** \( (1, 4) \) - **Point B:** \( (2, 2) \) - **Point C:** \( (3, 0) \) - **Point D:** \( (4, -2) \) After applying the vertical compression: - **Transformed Point A:** \( (1, 2) \) \( \left( \frac{1}{2} \times 4 = 2 \right) \) - **Transformed Point B:** \( (2, 1) \) \( \left( \frac{1}{2} \times 2 = 1 \right) \) - **Transformed Point C:** \( (3, 0) \) \( \left( \frac{1}{2} \times 0 = 0 \right) \) - **Transformed Point D:** \( (4, -1) \) \( \left( \frac{1}{2} \times (-2) = -1 \right) \) Plot these transformed points and connect them to form the graph of \( y = \frac{1}{2} f(x) \). ### Key Characteristics of \( y = \frac{1}{2} f(x) \): - **Vertical Compression:** All y-values are halved, making the graph "shorter" vertically. - **Shape Preservation:** The overall shape of the graph remains the same; only the vertical scale changes. - **Axis:** The x-axis remains the same since the transformation only affects the y-values. ### Example: If the original graph \( y = f(x) \) is a sine wave with amplitude 2, then \( y = \frac{1}{2} f(x) \) will be a sine wave with amplitude 1. The peaks and troughs are closer to the x-axis, but the frequency and period remain unchanged. --- **Tip:** Always ensure to apply the transformation to multiple points to maintain accuracy in the new graph. If possible, use graph paper or a graphing tool to assist in plotting the transformed points precisely.