Final answer:
To sketch the conic with the equation r=4/(1+3 cos θ), we identify that it's a hyperbola due to the coefficient greater than 1. We find the vertices at θ=0 and θ=π, which are (1, 0) and (-2, 0). The sketch involves drawing the two branches of the hyperbola and labeling these vertices.
Step-by-step explanation:
To sketch the conic represented by the polar equation r=4/(1+3 cos θ) and label the vertices, we need to recognize the type of conic it represents. Given the cosine in the denominator, this is a form of a polar equation of a conic with a focus at the origin.
First, we can identify the conic as an ellipse, parabola, or hyperbola based on the eccentricity given by the coefficient of the cos θ term. Since the coefficient is 3, which is greater than 1, we are dealing with a hyperbola. To find the vertices, we look for the θ values that make cos θ = 1 or -1 which are the maximum and minimum distances from the focus.
At θ = 0, cos θ = 1, so the distance, r, is minimum and the vertex is at (4/4, 0) or (1, 0). At θ = π, cos θ = -1, and thus the other vertex is at (4/(-2), 0) or (-2, 0). These vertices define the transverse axis of the hyperbola.
Now we can sketch the conic by plotting these vertices and then drawing the characteristic hyperbola shape around them, being careful to draw the two separate branches that go towards infinity. To complete the sketch, we label the vertices as (1, 0) and (-2, 0).