Question

The graph of lower r equals negative 14 cosine theta has which of the following characteristics?
(1 point)
Responses

circle, diameter of 14, center at left parenthesis negative 7 comma 0 right parenthesis
circle, diameter of 14, center at Image with alt text: left parenthesis negative 7 comma 0 right parenthesis

circle, diameter of 14, center at left parenthesis negative 14 comma 0 right parenthesis
circle, diameter of 14, center at Image with alt text: left parenthesis negative 14 comma 0 right parenthesis

circle, radius of 14, center at left parenthesis negative 7 comma 0 right parenthesis
circle, radius of 14, center at Image with alt text: left parenthesis negative 7 comma 0 right parenthesis

circle, radius of 14, center at left parenthesis negative 14 comma 0 right parenthesis

Answer 1

To analyze the graph of \( r = -14 \cos \theta \), we can convert this polar equation into Cartesian coordinates to determine the characteristics.

1. **Understanding the Equation:**
The equation \( r = -14 \cos \theta \) suggests that the polar graph is influenced by the cosine function, and the negative sign indicates a reflection over the origin.

2. **Identifying Features:**
Polar equations of the form \( r = a \cos \theta \) represent circles when \( a \) is positive. The general form \( r = a \cos \theta \) corresponds to a circle centered at \( (a/2, 0) \) with a radius of \( |a|/2 \).

3. **Radius and Center Calculations:**
Here, \( a = -14 \). Referring to the form, we can extract:
- The center can be identified at \( (-7, 0) \) because:
\[
\text{Center} = \left(-\frac{a}{2}, 0\right) = \left(-\frac{-14}{2}, 0\right) = (-7, 0)
\]
- The radius is \( |a|/2 \):
\[
\text{Radius} = \frac{|-14|}{2} = 7
\]

4. **Analyzing the Negative Radius:**
The negative radius implies a reflection over the origin. Thus, although the center is at \( (-7, 0) \), the graph ultimately still describes a circle.

Now, combining these characteristics:
- The center is at \( (-7, 0) \).
- The radius is \( 7 \), not \( 14 \).

Thus, none of the options perfectly match all the derived characteristics, especially in radius. The correct interpretation, if the options had included the radius as \( 7 \) rather than \( 14 \), would have been:
- Circle, radius of 7, center at \( (-7, 0) \).

However, based on the closest option possible based on description:
- If asked which is more accurate from the given options, we would have to state that none completely fit but recognize that it’s around the \( (-7, 0) \) center with the properties of a circle.

If one must choose among the closest options, the one that accurately describes the center would be:
**Circle, radius of 14, center at \( (-7, 0) \)** would be the misinterpreted statement that tries to identify the properties rounded to a certain figure.

Make sure to clarify any inconsistencies in the problem options next!