Agraph of \( y=f(x) \) is given. No formula for \( f \) is given. Find the graph of \( g(x)=f\left(-\frac{1}{2} x\right) \).

To find the graph of \( g(x) = f\left(-\frac{1}{2} x\right) \) based on the graph of \( y = f(x) \), we need to apply two transformations to the graph of \( f(x) \): 1. **Horizontal Reflection**: The term \( -\frac{1}{2} x \) indicates a reflection across the y-axis because of the negative sign. This means that for every point \( (a, b) \) on the graph of \( f(x) \), the corresponding point on the graph of \( g(x) \) will be \( (-a, b) \). 2. **Horizontal Stretch**: The factor of \( \frac{1}{2} \) indicates a horizontal stretch by a factor of 2. This means that for every point \( (a, b) \) on the graph of \( f(x) \), the corresponding point on the graph of \( g(x) \) will be \( (-2a, b) \). ### Steps to Transform the Graph: 1. **Reflect the graph of \( f(x) \)** across the y-axis. If a point \( (a, b) \) is on the graph of \( f(x) \), then the reflected point will be \( (-a, b) \). 2. **Stretch the reflected graph horizontally** by a factor of 2. This means that if you have a point \( (-a, b) \) after reflection, it will become \( (-2a, b) \). ### Summary of the Transformation: - Start with a point \( (a, f(a)) \) on the graph of \( f(x) \). - After reflection, this point becomes \( (-a, f(a)) \). - After the horizontal stretch, this point becomes \( (-2a, f(a)) \). ### Conclusion: To sketch the graph of \( g(x) = f\left(-\frac{1}{2} x\right) \), take each point on the graph of \( f(x) \), reflect it across the y-axis, and then stretch it horizontally by a factor of 2. This will give you the graph of \( g(x) \).