Problem 65 - College Trigonometry

Question

For Exercises \(65 a\) and 65 b below, let \(f(\theta)=\cos (\theta)\) and \(g(\theta)=2-\sin (\theta)\). (a) Using your graphing calculator, compare the graph of \(r=f(\theta)\) to each of the graphs of \(r=f\left(\theta+\frac{\pi}{4}\right), r=f\left(\theta+\frac{3 \pi}{4}\right), r=f\left(\theta-\frac{\pi}{4}\right)\) and \(r=f\left(\theta-\frac{3 \pi}{4}\right)\). Repeat this process for \(g(\theta)\). In general, how do you think the graph of \(r=f(\theta+\alpha)\) compares with the graph of \(r=f(\theta)\) ? (b) Using your graphing calculator, compare the graph of \(r=f(\theta)\) to each of the graphs of \(r=2 f(\theta), r=\frac{1}{2} f(\theta), r=-f(\theta)\) and \(r=-3 f(\theta)\). Repeat this process for \(g(\theta) .\) In general, how do you think the graph of \(r=k \cdot f(\theta)\) compares with the graph of \(r=f(\theta) ?\) (Does it matter if \(k>0\) or \(k<0 ?)\)

Step-by-Step Solution

Answer

Shifts rotate the graph; scaling stretches, shrinks, or reflects it.

Step 1: Understanding the Functions

The given functions are \( f(\theta) = \cos(\theta) \) and \( g(\theta) = 2 - \sin(\theta) \). We need to investigate how shifts and scalings affect their polar graphs.

Step 2: Exploring Horizontal Shifts

Use a graphing calculator to compare the polar graph of \( r = f(\theta) \) with shifted versions, i.e., \( r = f(\theta + \frac{\pi}{4}) \), \( r = f(\theta + \frac{3\pi}{4}) \), \( r = f(\theta - \frac{\pi}{4}) \), and \( r = f(\theta - \frac{3\pi}{4}) \). Notice how shifting \( \theta \) moves the graph around the polar coordinate system.

Step 3: Generalizing Horizontal Shifts

When comparing \( r = f(\theta + \alpha) \) to \( r = f(\theta) \), the graph experiences a rotation by \( \alpha \) radians counterclockwise if \( \alpha > 0 \), and clockwise if \( \alpha < 0 \). This effect is the same for both functions \( f(\theta) \) and \( g(\theta) \).

Step 4: Exploring Vertical Scaling

Use your calculator to compare \( r = f(\theta) \) with \( r = 2f(\theta) \), \( r = \frac{1}{2}f(\theta) \), \( r = -f(\theta) \), and \( r = -3f(\theta) \). Observe how multiplying by a constant \( k \) affects the size or orientation of the graph.

Step 5: Understanding Scaling Effects

The graph of \( r = k\cdot f(\theta) \) stretches the graph if \(|k| > 1\), and shrinks it if \(|k| < 1\). If \(k < 0\), the graph is also reflected across the polar axis. This behavior holds for both \( f(\theta) \) and \( g(\theta) \).

Step 6: Conclusions

In general, \( r = f(\theta + \alpha) \) rotates the graph, while \( r = k\cdot f(\theta) \) scales and possibly reflects it depending on \( k \). These transformations have consistent effects on polar equations of the form \( f(\theta) \) and \( g(\theta) \).

Key Concepts

Graph Transformations

Graph transformations in polar coordinates involve changing the way a graph is displayed on the polar grid. A crucial aspect of these transformations includes translating or rotating the graph and adjusting its size or orientation. Understanding these transformations helps in visualizing how the graph's appearance changes without altering the function itself.

• Rotation: This is accomplished by shifting the angle \( \theta \). For example, changing \( r = f(\theta) \) to \( r = f(\theta + \alpha) \) effectively rotates the graph by \( \alpha \) radians. Positive \( \alpha \) leads to counterclockwise rotation, and negative \( \alpha \) results in clockwise rotation.
• Reflection and Scaling: These include flipping the graph or changing its size, achieved via multiplication by a constant, \( k \). Reflection occurs when \( k \) is negative, while scaling changes the graph's size proportionally to \(|k|\).

Through graph transformations, abstract equations can be visualized more clearly, making the underlying mathematical concepts easier to grasp.

Trigonometric Functions

Trigonometric functions like \( \cos(\theta) \) and \( \sin(\theta) \) serve as the foundation for many polar graphs. These functions describe periodic phenomena, with regular oscillations that are perfectly suited to polar coordinates.

• Cosine Function: Defined as \( f(\theta) = \cos(\theta) \), it exhibits even symmetry, creating symmetric polar graph structures like circles or loops along the radial direction.
• Sine Function Variations: Alterations, such as \( g(\theta) = 2 - \sin(\theta) \), shift the basic sine graph up or down (this is covered under function shifts). The sine function induces radial oscillations at consistent intervals around the circle.

Understanding these functions highlights how their properties translate into geometric shapes within the polar coordinate system.

Function Shifts

Function shifts occur when constants are added to or subtracted from the angle \( \theta \) in trigonometric functions. This results in rotating the graph of the function along the polar coordinate system. Observing these shifts can show how much rotation is required to bring a graph from one position to another.

• Horizontal Shifts: Directly affect the angle \( \theta \), represented as \( \theta + \alpha \). The resulting graph rotates without changing shape, providing insight into how the original polar pattern shifts around the origin.
• Dynamic Graph Behavior: Through successive graphing operations, these shifts reveal the inherent symmetry or repeated patterns, particularly useful in problems involving periodical behavior.

In polar graphs, function shifts significantly impact orientation, furthering comprehension of periodic motion and oscillations.

Polar Graph Scaling

Polar graph scaling involves changing the size or direction of a graph relative to the polar origin. This technique is particularly useful when seeking to visualize the impact of varying scales on polar equations.

• Vertical Scaling: Involves altering the radial distance by multiplying \( f(\theta) \) by a constant \( k \). If \(|k| > 1\), the graph expands; if \(|k| < 1\), it contracts.
• Reflection: Occurs when \( k \) is negative. In this case, the graph is inverted across the horizontal polar axis, essentially flipping its orientation.

Scaling provides valuable insight into the flexibility of polar equations, allowing for adjustments that reveal different aspects of the graphs' structure and symmetry.