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Using another polar coordinate grid, graph the function r(θ) = 4 + 2sinθ. What difference(s) do you notice between this graph and the one in the previous problem? How could we anticipate these differences just by looking at the formula for each function?

Using another polar coordinate grid, graph the function r(θ) = 4 + 2sinθ. What difference(s) do you notice between this graph and the one in the previous problem? How could we anticipate these differences just by looking at the formula for each function?

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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7. Using another polar coordinate grid, graph the function r(θ) = 4 + 2sinθ. What difference(s) do you notice between this graph and the one in the previous problem? How could we anticipate these differences just by looking at the formula for each function? The second picture being the previous problem it was talking about
**Graph of the Function \( r(\theta) = 2 + 4\cos\theta \) in Rectangular Coordinates** This image displays the graph of the function \( r(\theta) = 2 + 4\cos\theta \) using rectangular coordinates. **Graph Description:** - **Axes:** - The horizontal axis represents \( \theta \), which ranges from \( 0 \) to \( 2\pi \). - The vertical axis represents \( r \). - **Key Points:** - The point at \( \theta = 0 \) is marked at \( (0, 6) \). - At \( \theta = \pi \), the graph reaches its minimum value, marked at \( (\pi, -2) \). - At \( \theta = 2\pi \), the graph returns to the maximum, marked at \( (2\pi, 6) \). - **Shape and Behavior:** - The graph appears as a sinusoidal wave. - It starts at its maximum value of 6 at \( \theta = 0 \), descends to the minimum value of -2 at \( \theta = \pi \), and returns to the maximum value of 6 at \( \theta = 2\pi \). - The pattern is symmetrical around \( \theta = \pi \). This graph provides a visualization of how the function \( r(\theta) = 2 + 4\cos\theta \) behaves, showcasing its periodic nature and the influence of the cosine component in shaping the curve. The function's oscillation between maximum and minimum values is clearly illustrated.
Transcribed Image Text:**Graph of the Function \( r(\theta) = 2 + 4\cos\theta \) in Rectangular Coordinates** This image displays the graph of the function \( r(\theta) = 2 + 4\cos\theta \) using rectangular coordinates. **Graph Description:** - **Axes:** - The horizontal axis represents \( \theta \), which ranges from \( 0 \) to \( 2\pi \). - The vertical axis represents \( r \). - **Key Points:** - The point at \( \theta = 0 \) is marked at \( (0, 6) \). - At \( \theta = \pi \), the graph reaches its minimum value, marked at \( (\pi, -2) \). - At \( \theta = 2\pi \), the graph returns to the maximum, marked at \( (2\pi, 6) \). - **Shape and Behavior:** - The graph appears as a sinusoidal wave. - It starts at its maximum value of 6 at \( \theta = 0 \), descends to the minimum value of -2 at \( \theta = \pi \), and returns to the maximum value of 6 at \( \theta = 2\pi \). - The pattern is symmetrical around \( \theta = \pi \). This graph provides a visualization of how the function \( r(\theta) = 2 + 4\cos\theta \) behaves, showcasing its periodic nature and the influence of the cosine component in shaping the curve. The function's oscillation between maximum and minimum values is clearly illustrated.
**Transcription and Explanation for Educational Website:** **Polar Coordinate Graphing Task:** 1. **Task Description:** - **Problem 7:** Using another polar coordinate grid, graph the function \( r(\theta) = 4 + 2\sin\theta \). - **Discussion Points:** - What difference(s) do you notice between this graph and the one in the previous problem? - How could we anticipate these differences just by looking at the formula for each function? 2. **Graph Description:** - The diagram features a polar coordinate grid, which is circular and divided into degrees starting from \( \theta = 0^\circ \) and progressing counterclockwise by increments of \( 15^\circ \) all the way through \( 360^\circ \). - The radial lines emanate outward from the center and are marked at angles such as \( 0^\circ \), \( 30^\circ \), \( 60^\circ \), and so on. - Concentric circles are drawn to represent different radius lengths which increase outwardly from the center of the grid. 3. **Analyzing the Function:** - The function \( r(\theta) = 4 + 2\sin\theta \) combines a constant \( 4 \) with a sinusoidal component \( 2\sin\theta \), indicating a graph that will have variability in the radius based on the angle \(\theta\). - Differences from previous graphs can be understood by examining the change in amplitude (indicated by the coefficient of \(\sin\theta\)) and the overall shift in the radius caused by the constant term \(4\). 4. **Anticipating Graph Characteristics:** - By analyzing the formula: - The constant term \(4\) suggests a minimum radius of 4, creating a baseline circular form. - The amplitude of \(2\) in the sinusoidal term \(2\sin\theta\) indicates that the radius will fluctuate by 2 units above and below the base circle, resulting in a pattern influenced by the sinusoidal wave. - This typically results in an oscillating petal-like pattern around the circle. This explanation serves to assist in understanding the dynamic nature of sinusoidal graphs in polar coordinates and their varied appearances based on adjustments within the function.
Transcribed Image Text:**Transcription and Explanation for Educational Website:** **Polar Coordinate Graphing Task:** 1. **Task Description:** - **Problem 7:** Using another polar coordinate grid, graph the function \( r(\theta) = 4 + 2\sin\theta \). - **Discussion Points:** - What difference(s) do you notice between this graph and the one in the previous problem? - How could we anticipate these differences just by looking at the formula for each function? 2. **Graph Description:** - The diagram features a polar coordinate grid, which is circular and divided into degrees starting from \( \theta = 0^\circ \) and progressing counterclockwise by increments of \( 15^\circ \) all the way through \( 360^\circ \). - The radial lines emanate outward from the center and are marked at angles such as \( 0^\circ \), \( 30^\circ \), \( 60^\circ \), and so on. - Concentric circles are drawn to represent different radius lengths which increase outwardly from the center of the grid. 3. **Analyzing the Function:** - The function \( r(\theta) = 4 + 2\sin\theta \) combines a constant \( 4 \) with a sinusoidal component \( 2\sin\theta \), indicating a graph that will have variability in the radius based on the angle \(\theta\). - Differences from previous graphs can be understood by examining the change in amplitude (indicated by the coefficient of \(\sin\theta\)) and the overall shift in the radius caused by the constant term \(4\). 4. **Anticipating Graph Characteristics:** - By analyzing the formula: - The constant term \(4\) suggests a minimum radius of 4, creating a baseline circular form. - The amplitude of \(2\) in the sinusoidal term \(2\sin\theta\) indicates that the radius will fluctuate by 2 units above and below the base circle, resulting in a pattern influenced by the sinusoidal wave. - This typically results in an oscillating petal-like pattern around the circle. This explanation serves to assist in understanding the dynamic nature of sinusoidal graphs in polar coordinates and their varied appearances based on adjustments within the function.
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