Question

Graph the following spirals. Indicate the direction in which the spiral is generated as $\theta$ increases, where $\theta >0.$ Let $a=1$ and $a=-1$ Spiral of Archimedes: $r=a\theta$

Short answer

Answer: For $a=1$, the spiral moves in a counterclockwise direction as θ increases. For $a=-1$, the spiral moves in a clockwise direction as θ increases.

Step-by-step solution

Understanding the equation

The given equation is in polar coordinates, where r represents the radial distance from the origin, and θ represents the angle. The equation is $r=a\theta$. Here, a is a constant, and we have to consider two cases: $a=1$ and $a=-1$.

Set values for θ

In order to visualize the spiral, we need to set values for θ. Since we want to graph the spiral in both θ increasing and decreasing directions, we'll choose an interval for θ from $0$ to $4\pi$. This should give us a couple of loops around the origin.

Graphing for $a=1$

With $a=1$, the equation becomes $r=\theta$. We'll plot points for different values of θ in the interval $0$ to $4\pi$. As we do this, we will notice that as θ increases, the spiral moves outwards from the origin in a counterclockwise direction. This is because r is also increasing as θ increases.

Graphing for $a=-1$

With $a=-1$, the equation becomes $r=-\theta$. We'll plot points for different values of θ in the interval $0$ to $4\pi$. As we do this, we will notice that as θ increases, the spiral moves outwards from the origin, but in a clockwise direction. This is due to the negative value of a, which causes r to have the opposite sign of θ.

Combining the Graphs

Now that we have graphs for both cases of a (a = 1 and a = -1), we need to combine them to display on the same plot. This will show us the difference in the direction of the spirals as θ increases.

Indicate the Direction

Since we are asked to indicate the direction in which the spiral is generated as θ increases, we'll use arrows on the graph to show the direction for each case. For $a=1$, the arrow should point counterclockwise, while for $a=-1$, the arrow should point clockwise. In conclusion, we have successfully graphed the Spiral of Archimedes for both $a=1$ and $a=-1$. We have also determined that the spiral moves in counterclockwise direction for $a=1$ and clockwise direction for $a=-1$ as θ increases.

Key concepts

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system that specifies a point's position by its distance from the origin (or pole) and its direction given by an angle. Unlike the Cartesian coordinate system, which uses x and y axes, polar coordinates use the radial distance, denoted as $r$, and the angle, denoted as $\theta$, from a fixed direction (usually the positive x-axis). This system is particularly useful in scenarios where rotation or circular movement is involved, such as graphing spirals.

• $r$ is the distance from the origin to the point.
• $\theta$ is the angle measured in radians from the positive x-axis.

The relationship between polar and Cartesian coordinates can be defined with the formulas:

• $x=r\cos \theta$
• $y=r\sin \theta$

Radial Distance

Radial distance, represented by $r$ in polar coordinates, is the straight-line distance between the origin (pole) and a specific point. It plays a central role in understanding how spirals like the Spiral of Archimedes are plotted, as this distance changes linearly with the angle $\theta$.

In the equation for the Spiral of Archimedes, $r=a\theta$, the radial distance changes as $\theta$ varies. Here, $a$ is a constant that influences how tight or loose the spiral is. When $a=1$, the radial distance increases steadily with $\theta$ resulting in a typical outward spiral. Conversely, when $a=-1$, the spiral moves inwards, flipping the direction. The sign and value of $a$ are crucial to understanding how the spiral behaves.

• Positive $a$: Spiral expands counterclockwise as $\theta$ increases.
• Negative $a$: Spiral expands clockwise as $\theta$ increases.

Graphing Spirals

Graphing spirals involves plotting points in polar coordinates where the radial distance $r$ is a function of the angle $\theta$. For the Spiral of Archimedes, $r=a\theta$, we plot points over a range of $\theta$ values, thus visually presenting the path of the spiral.

• Start at $\theta =0$ where $r=0$.
• Increase $\theta$ incrementally, plotting the corresponding points.

By observing the trend of plotted points, one can visualize whose spiral expands outwards from the origin. As $\theta$ increases, and for $a=1$, the points create a counterclockwise outward spiral. For $a=-1$, the spiral instead winds outward in a clockwise direction. This difference in directionality is greatly influenced by the sign of $a$.

Increasing $\theta$ over an interval such as $0$ to $4\pi$ helps ensure the spiral makes multiple loops, providing a fuller view of the structure.

Directionality in Polar Graphs

Directionality refers to the way a graph is oriented or unfolds as $\theta$ changes, specifically in polar coordinates. For the Spiral of Archimedes, determining whether the spiral expands in a clockwise or counterclockwise direction depends on the constant $a$ in the equation $r=a\theta$.

When exploring the graph, the spiral’s direction is visualized as follows:

• For $a=1$, the spiral’s radial distance increases with $\theta$, making it expand counterclockwise from the origin.
• For $a=-1$, the spiral expands clockwise, as the negative sign reverses the direction of the increasing $\theta$.

Arrows can be drawn on the graph to indicate these directions for clarity. By observing the path determined by changing $\theta$, one can distinguish the spiral’s flow based on these indications, vividly illustrating directionality through a visual medium.