Problem 73 - Precalculus with Limits

Question

The graph of \(r=f(\theta)\) is rotated about the pole through an angle \(\theta\). Show that the equation of the rotated graph \(r=f(\theta-\phi)\).

Step-by-Step Solution

Answer

When the graph of a polar function \(r=f(\theta)\) is rotated by an angle \(\phi\), the equation of the graph changes to \(r=f(\(\theta-\phi\))\), demonstrating the rotational nature of polar coordinates.

Step 1: Understanding Rotation in Polar Coordinates

In polar coordinates, a point is defined by two parameters: the distance \(r\) from the origin (pole) and the angle \(\theta\) from the x-axis. When the graph of a function \(r=f(\theta)\) is rotated by an angle \(\phi\), the points on the graph maintain their distance \(r\) from the pole, but their angle is increased by \(\phi\).

Step 2: Describe the Transformation

To describe this transformation algebraically, it's recognized that in the new graph, the distance \(r\) from a point to the origin still equals the output of the function \(f\), but the input angle has been shifted by \(\phi\). This leads to the transformed function being \(r=f(\(\theta-\phi\))\). This means for a given \(\theta\) in the new graph, the corresponding \(\theta\) in the original graph was \(\theta-\phi\).

Step 3: Understanding the Implication

This transformation thus doesn't change the distances of points from the pole, nor does it change the 'shape' of the graph - it only 'rotates' it around the pole. Basically, each point on the rotated graph corresponds to some point on the original graph, but at a \(\phi\) greater angle. So the new equation for the rotated graph becomes \(r=f(\(\theta-\phi\)).

Key Concepts

Polar Graph Rotation

When we talk about rotating a graph in polar coordinates, we're discussing a shift that occurs while keeping the graph's shape intact. In polar coordinates, everything revolves around angles and distance. Imagine fixing your focus on a point, the pole, and describing an object based on how far it is from you (the "r" value) and the angle ("\(\theta\)") it makes with a reference direction, usually the x-axis.

Rotating a polar graph by an angle \(\phi\) means all points on the graph move along a circular path around the pole. The key is that the radius (r) doesn’t change during this rotation, it’s only the angle \(\theta\) that is adjusted. If you started with a function like \(r = f(\theta)\) and want to rotate it, each angle would increase by \(\phi\).

Algebraically, this operation adjusts the input of the function: you use \(\theta - \phi\) when plugging into \(f\). So, the rotated graph has the equation \(r = f(\theta - \phi)\). This simple change in the formula captures the concept of rotating the graph around the origin without stretching or squishing it.

Function Transformation

Function transformation involves shifting, stretching, or reflecting the original graph in various ways. For polar coordinates, we specifically focus on shifting, in the form of a rotation. Unlike Cartesian coordinates, where transformations often mean changes in position or size, in polar coordinates, the transformation we discussed is purely angular.

By adjusting the "input" angle for our function from \(\theta\) to \(\theta - \phi\), we effectively "shift" every angle in the original function. It's like when you adjust your watch to account for daylight saving time, moving the time forward or backward—except here, we're shifting angles. This angular transformation gives rise to a new perspective of the original shape, and is a precise shift, keeping the distances from the origin, or pole, constant.

This kind of transformation highlights the elegance of polar graphs: a simple algebraic change reflects a physical rotation, showing the interconnectedness of angles and geometric shapes in polar systems.

Distance from the Origin

In polar coordinates, "distance from the origin" is a fundamental concept. The distance, denoted as \(r\), measures how far a point on the graph is from the origin, or the pole. Despite any transformations applied to a polar graph, such as rotations, this radial distance remains unchanged.

While discussing polar graph rotation, it's crucial to understand that the distance \(r\) tells us how "stretched" the graph is from the center outwards. Each point's position is therefore described fully by its distance \(r\) and its angle \(\theta\).

Maintaining this distance means that any transformations such as rotations only involve angular adjustments. No matter how the graph turns around the origin, the radial lines connecting each point to the center stay the same in length. Thus, during a rotation, the graph maintains its overall size and shape, not deviating from the original distances laid out by the function \(f(\theta)\).

Understanding this concept is key for visualizing and working with graphs in polar coordinates, providing stability in the image as angles shift.