Let r f be a polar function where f 5 The (2024)

#### Solution By Steps***Step 1: Analyzing the Given Polar Function***Given $r=f(\theta)=5$, this represents a circle with radius $5$ centered at the origin in the polar coordinate system.***Step 2: Understanding the Graph***The graph of $r=f(\theta)$ for $0 \leq \theta \leq \pi$ is a semicircle in the first and second quadrants.***Step 3: Evaluating the Given Options***(A) $r=g(\theta)=-5$ for $0 \leq \theta \leq \pi$ represents a circle with radius $5$ but centered at the origin with the same shape as the original semicircle.(B) $r=h(\theta)=\frac{1}{5}$ for $0 \leq \theta \leq \pi$ represents a circle with radius $\frac{1}{5}$, not the same as the original semicircle.(C) $r=k(\theta)=-5$ for $\pi \leq \theta \leq 2\pi$ represents a circle with radius $5$ but centered at the origin, not the same as the original semicircle.(D) $r=m(\theta)=\frac{1}{5}$ for $\pi \leq \theta \leq 2\pi$ represents a circle with radius $\frac{1}{5}$, not the same as the original semicircle.#### Final AnswerThe graph of the polar function $r=g(\theta)=-5$ for $0 \leq \theta \leq \pi$ is the same semicircle as $r=f(\theta)=5$ for $0 \leq \theta \leq \pi$.#### Key ConceptPolar Function Symmetry#### Key Concept ExplanationUnderstanding the symmetry of polar functions helps identify shapes like circles and semicircles in the polar coordinate system. The sign change in the radial function $r$ reflects a reflection across the origin, preserving the shape but changing the orientation.