Question

Explain how the graph of each conic differs from the graph of \(r=\frac{4}{1+\sin \theta} .\) (a) \(r=\frac{4}{1-\cos \theta}\) (b) \(r=\frac{4}{1-\sin \theta}\) (c) \(r=\frac{4}{1+\cos \theta}\) (d) \(r=\frac{4}{1-\sin (\theta-\pi / 4)}\)

Short answer

The different conic sections expressed in polar coordinates have different graph representations, with specific transformations such as reflection and rotation when the equation changes. The comparison between given polar coordinate and the presented ones revealed different transformations like mirroring about the x-axis or certain lines, and rotations by different degrees.

Step-by-step solution

Graph for (a)

The equation for (a) is \(r=\frac{4}{1-\cos \theta}\). Graph this equation and compare it to the given equation. Note the differences. Here, the effect of changing \(\sin \theta\) to \(-\cos \theta\) in the denominator mirrors the graph about the line \(\theta=\pi/4\)

Graph for (b)

The equation for (b) is \(r=\frac{4}{1-\sin \theta}\). Graph this equation and compare it to the given equation. The function is mirrored about the x-axis. This happens because the denominator of \(r\) changes from \(1+\sin \theta\) to \(1-\sin \theta\)

Graph for (c)

The equation for (c) is \(r=\frac{4}{1+\cos \theta}\). Graph this equation and compare it to the given equation. Here, the effect of changing \(\sin \theta\) to \(\cos \theta\) in the denominator rotates the graph through 90 degrees counterclockwise.

Graph for (d)

The equation for (d) is \(r=\frac{4}{1-\sin (\theta-\pi / 4)}\). Graph this equation and compare with given equation. The effect of replacing \(\theta\) by \(\theta-\pi / 4\) in the formula for \(r\) rotates the graph through 45 degrees clockwise.

Key concepts

Conic Sections in Polar Coordinates

When we start to explore conic sections in polar coordinates, we embark on a journey that combines geometric intuition with algebraic manipulation. To understand this concept, picture the conics - circles, ellipses, hyperbolas, and parabolas - as shapes produced by slicing a double cone. In a polar coordinate system, instead of using the familiar 'x' and 'y' of Cartesian coordinates, we identify points using a distance from the origin, denoted as 'r', and an angle from the positive x-axis, which we label as '\(\theta\)' (theta).

This choice of representation is particularly advantageous when dealing with conics. For instance, the polar equation \(r=\frac{e}{1+e\cos\theta}\) represents a conic section, where 'e' is the eccentricity. If 'e' is less than 1, we get an ellipse, equal to 1 a parabola, and greater than 1 a hyperbola. The exercise at hand showcases variations of a conic section equation where different alterations to the denominator transform the graph into diverse shapes, each reflecting a property of conic sections in polar coordinates.

Understanding how the graph changes when we alter the equation helps to visualize these abstract concepts. Intuitively, changes in the angle component '\(\theta\)' lead to rotations, while modifications in 'r' affect the size and orientation. By learning to graph these equations, students visualize the essence of conics in a dynamic polar setting.

Transformation of Graphs

The transformation of graphs is a crucial skill in mathematics, allowing students to predict and understand changes in a visual representation of an equation without plotting countless points. Transformations can include translations, reflections, stretches, and rotations. Looking at the equation \(r=\frac{4}{1+\sin\theta}\) and its variations, as provided in the exercise, we can analyze such transformations.

For instance, a simple change from '+\sin\theta' to '-\cos\theta' results in mirroring the graph about the line '\(\theta=\frac{\pi}{4}\)'. This kind of reflection is a transformation where the graph flips over a specified line, changing its orientation. Similarly, replacing a '+\sin\theta' with a '-\sin\theta' mirrors the graph about the horizontal axis, essentially flipping it upside down. Understanding these fundamental transformations allows students to grasp the effects these operations have on polar graphs - equipping them with the tools to manipulate and understand complex equations more easily.

The skill to visualize and perform graph transformations saves time and elevates cognitive comprehension. By mastering transformations, students can efficiently sketch the curves of more elaborate equations, which is an essential skill in higher mathematics and physics.

Polar Equations

Polar equations define relationships between the radius 'r' and the angle '\(\theta\)' in a polar coordinate system. They can describe spirals, circles, and various other curves that may be complicated to express in traditional Cartesian coordinates. A fundamental form of a polar equation is \(r=f(\theta)\), where 'r' is a function of '\(\theta\)'.

In the exercise at hand, variations of \(r=\frac{4}{1+\sin\theta}\) are used to exhibit how polar equations describe conic sections. A small tweak to the polar equation results in a significant transformation of its graph. For example, changing '\(\sin\theta\)' to '\(\cos\theta\)' not only alters the graph's appearance but can also shift it by 90 degrees. Furthermore, introducing an angle change like '\(\theta-\frac{\pi}{4}\)' in the denominator of part (d) corresponds to rotation of the graph by 45 degrees clockwise.

The efficacy of polar equations becomes evident when dealing with angles and lengths from a fixed point, providing a straightforward method to describe curves that are symmetrical about the origin or a particular line. Mastery of polar equations and their transformations paves the way for students to deal with complex curves and prepares them for advanced topics in calculus, physics, and engineering.