Question
Describe the vertical asymptote(s) and hole(s) for the graph of y=(x−2)(x+5)(x+5)(x+3) (1 point) Responses asymptote: x = - 3 and hole: x = - 5 asymptote: x = - 3 and hole: x = - 5 asymptotes: x = - 3 and x = - 5 asymptotes: x = - 3 and x = - 5 asymptote: x = - 2 and hole: x = -3 asymptote: x = - 2 and hole: x = -3 asymptote: x = 3 and hole: x = 5
Answers
Answered by
GPT-4o mini
To analyze the function \( y = (x - 2)(x + 5)(x + 5)(x + 3) \), we first need to look for any vertical asymptotes and holes.
**Vertical Asymptotes:**
Vertical asymptotes occur in rational functions where the denominator is zero but the numerator is not zero at that point. However, the given function is a polynomial, not in a fractional form that would allow for vertical asymptotes. Since the function doesn't have a denominator, there are **no vertical asymptotes**.
**Holes:**
Holes in the graph occur where a factor cancels in a rational function. In this case, since the function is a polynomial with no denominators, there are also **no holes**.
From the given information, the keywords that best clarify vertical asymptotes and holes are that both are absent in the polynomial function:
Thus, the correct response would be that **there are no vertical asymptotes and no holes**. However, none of the provided choices exactly state that, which might indicate an error or a misunderstanding in the options given. If strictly speaking based on the provided options, none of them accurately reflect the characteristics of the function provided.
**Vertical Asymptotes:**
Vertical asymptotes occur in rational functions where the denominator is zero but the numerator is not zero at that point. However, the given function is a polynomial, not in a fractional form that would allow for vertical asymptotes. Since the function doesn't have a denominator, there are **no vertical asymptotes**.
**Holes:**
Holes in the graph occur where a factor cancels in a rational function. In this case, since the function is a polynomial with no denominators, there are also **no holes**.
From the given information, the keywords that best clarify vertical asymptotes and holes are that both are absent in the polynomial function:
Thus, the correct response would be that **there are no vertical asymptotes and no holes**. However, none of the provided choices exactly state that, which might indicate an error or a misunderstanding in the options given. If strictly speaking based on the provided options, none of them accurately reflect the characteristics of the function provided.