What type of limacon is graphed by the polar equation r=4+2 sin Θ? Identify the axis of symmetry and horizontal and vertical intercepts.
Well, this limacon is definitely not your average circus performer! It belongs to the "Funhouse Limacon" family.
Now, when it comes to its graph, the axis of symmetry is simply the x-axis, because this limacon doesn't want to be asymmetrical, it wants to show off its perfectly balanced curves!
As for the horizontal and vertical intercepts, this limacon loves to make an entrance. The vertical intercept occurs when r equals zero, so when Θ is equal to zero or π. And the horizontal intercepts occur when r equals zero again, but this time when Θ is equal to π/2 or 3π/2.
So, remember to bring your clown nose and funny bone when admiring this limacon's graph. It's sure to put a smile on your face, or maybe even a honk!
The given polar equation is r = 4 + 2sin(Θ).
To determine the type of limacon, we can rewrite the equation in terms of r and Θ:
r = 4 + 2sin(Θ)
r - 4 = 2sin(Θ)
Let's analyze the equation's components:
1. Axis of Symmetry:
The axis of symmetry for a limacon is given by the polar equation r = a + bsin(Θ), where a and b are constants. In this case, a = 4 and b = 2. The axis of symmetry is the line with polar equation r = a, which means the axis of symmetry is r = 4.
2. Horizontal Intercept:
To find the horizontal intercept(s), we set r equal to zero:
0 = 4 + 2sin(Θ)
-2sin(Θ) = 4
sin(Θ) = -2/2
sin(Θ) = -1
Since the range of sin(Θ) is -1 to 1, there are no values of Θ that satisfy sin(Θ) = -1. Thus, the limacon does not intersect the horizontal axis.
3. Vertical Intercept:
To find the vertical intercept(s), we set Θ equal to zero:
r = 4 + 2sin(0)
r = 4 + 2(0)
r = 4
Therefore, the vertical intercept is at (4, 0).
To summarize:
- The given polar equation represents a limacon.
- The axis of symmetry is r = 4.
- The limacon does not intersect the horizontal axis.
- The vertical intercept is at (4, 0).
To understand the type of limacon graphed by the polar equation r = 4 + 2sin(Θ), let's break it down step by step:
1. A limacon is a type of polar graph that represents a curve with a single loop. It can have various forms, such as cardioids, loops, or combinations of different shapes.
2. The equation r = 4 + 2sin(Θ) indicates that the distance from the origin (r) is determined by the angle (Θ) and a sinusoidal function (2sin(Θ)). Since the coefficient of sin(Θ) is positive, the shape of the limacon will be looped inward towards the pole.
3. To identify the axis of symmetry, we need to examine the equation. Since Θ does not appear in the equation, it implies that the limacon has radial symmetry. In other words, the axis of symmetry is the polar axis (the horizontal line, r = 0).
4. To find the horizontal intercept(s), we set r = 0 in the equation. Solving for Θ gives 0 = 4 + 2sin(Θ). Since sin(Θ) can only range from -1 to 1, it's clear that there are no values of sin(Θ) that would make the equation true. Therefore, there are no horizontal intercepts.
5. To find the vertical intercept(s), we set Θ = 0 or π in the equation. For Θ = 0, r = 4 + 2sin(0) = 4 + 0 = 4. Therefore, the limacon intersects the vertical line r = 4 when Θ = 0. Similarly, when Θ = π, r = 4 + 2sin(π) = 4 - 2 = 2. This indicates that the limacon also intersects the vertical line r = 2 at Θ = π.
In summary, the polar equation r = 4 + 2sin(Θ) represents a limacon with radial symmetry, looped inward towards the pole. It has no horizontal intercepts but intersects the vertical lines r = 4 and r = 2 at Θ = 0 and Θ = π respectively.