part b
b. use the graph to describe the end behavior of the function.

a) as ( x o -infty ), ( y o -infty ) and as ( x o infty ), ( y o -infty )
b) as ( x o -infty ), ( y o -infty ) and as ( x o infty ), ( y o infty )
c) as ( x o -infty ), ( y o infty ) and as ( x o infty ), ( y o -infty )
d) as ( x o -infty ), ( y o infty ) and as ( x o infty ), ( y o infty )

Question

part b
b. use the graph to describe the end behavior of the function.

a) as ( x 	o -infty ), ( y 	o -infty ) and as ( x 	o infty ), ( y 	o -infty )
  b) as ( x 	o -infty ), ( y 	o -infty ) and as ( x 	o infty ), ( y 	o infty )
  c) as ( x 	o -infty ), ( y 	o infty ) and as ( x 	o infty ), ( y 	o -infty )
  d) as ( x 	o -infty ), ( y 	o infty ) and as ( x 	o infty ), ( y 	o infty )

Accepted Answer

# Brief Explanations:
To determine the end - behavior of a function from its graph (or related information, like the presence of a relative minimum at $x = 0,y=- 5$ and no relative maximum), we consider the leading term of the function (if it's a polynomial) or the general trend as $x$ approaches $\pm\infty$.

For a function with a relative minimum and the end - behavior options given, we analyze each option:

- Option A: As $x
ightarrow-\infty,y
ightarrow-\infty$ and as $x
ightarrow\infty,y
ightarrow-\infty$. This would be the case for a downward - opening parabola (degree 2, leading coefficient negative) or a function with an even degree and negative leading coefficient. But if there is a relative minimum, a function with both ends going to $-\infty$ (like a downward - opening parabola) has a maximum, not a minimum. So A is incorrect.
- Option B: As $x
ightarrow-\infty,y
ightarrow-\infty$ and as $x
ightarrow\infty,y
ightarrow\infty$. This is the behavior of a function with an odd degree and positive leading coefficient, but the left - hand end going to $-\infty$ and right - hand end going to $\infty$ for an odd - degree function with positive leading coefficient. However, if there is a relative minimum, let's check the other options.
- Option C: As $x
ightarrow-\infty,y
ightarrow\infty$ and as $x
ightarrow\infty,y
ightarrow-\infty$. This is the behavior of a function with an odd degree and negative leading coefficient. A function with odd degree and negative leading coefficient has a left - hand end going to $\infty$ (as $x
ightarrow-\infty$, $y = a(-\infty)^n$, $n$ odd, $a\lt0$ gives $y
ightarrow\infty$) and right - hand end going to $-\infty$ (as $x
ightarrow\infty$, $y=a(\infty)^n$, $n$ odd, $a\lt0$ gives $y
ightarrow-\infty$). If there is a relative minimum (which is possible for an odd - degree function with negative leading coefficient, as the function is decreasing then increasing then decreasing or other combinations), this can fit.
- Option D: As $x
ightarrow-\infty,y
ightarrow\infty$ and as $x
ightarrow\infty,y
ightarrow\infty$. This is the behavior of a function with an even degree and positive leading coefficient, which has a minimum (if the leading coefficient is positive) but both ends go to $\infty$. But we have a relative minimum at $x = 0,y=-5$, but if both ends go to $\infty$, the function would have a minimum, but let's check the other conditions. However, from the relative maximum being "none" and relative minimum at $x = 0$, if we consider the end - behavior, for a function with a relative minimum and the options, the correct end - behavior is when as $x
ightarrow-\infty,y
ightarrow\infty$ and as $x
ightarrow\infty,y
ightarrow-\infty$ (Option C). Wait, no, let's re - evaluate. Wait, maybe the function is a cubic function with a negative leading coefficient. A cubic function $y=ax^{3}+bx^{2}+cx + d$ with $a\lt0$ has end - behavior: as $x
ightarrow-\infty$, $y = a(-\infty)^{3}+...
ightarrow\infty$ (since $a\lt0$ and $(-\infty)^3=-\infty$, so $a\times(-\infty)= \infty$ when $a\lt0$) and as $x
ightarrow\infty$, $y=a(\infty)^{3}+...
ightarrow-\infty$ (since $a\lt0$ and $\infty^3=\infty$, so $a\times\infty = -\infty$ when $a\lt0$). And a cubic function can have a relative minimum and no relative maximum (if the derivative has only one real root, but usually a cubic has two critical points, one max and one min. Wait, maybe the graph is of a function like $y=-x^{3}-5$. The derivative $y'=-3x^{2}$, which has a critical point at $x = 0$. The second derivative $y''=-6x$, at $x = 0$, $y'' = 0$, but the function $y=-x^{3}-5$ has a point of inflection at $x = 0$, not a relative minimum. Wait, maybe the original function is a quadratic? No, a quadratic has two ends going in the same direction. Wait, maybe the user made a mistake in the relative minimum part, but based on the options, the correct end - behavior for a function with a relative minimum (assuming the function is a cubic with negative leading coefficient, even though a cubic with negative leading coefficient has a relative max and min, but if we ignore that for a second) or maybe the graph is of a function where as $x
ightarrow-\infty$, $y
ightarrow\infty$ and as $x
ightarrow\infty$, $y
ightarrow-\infty$, which is Option C. Wait, no, let's check the options again.

Wait, the key is to recall that for end - behavior:

- For even - degree polynomials:
    - If the leading coefficient is positive: as $x
ightarrow\pm\infty$, $y
ightarrow\infty$
    - If the leading coefficient is negative: as $x
ightarrow\pm\infty$, $y
ightarrow-\infty$
- For odd - degree polynomials:
    - If the leading coefficient is positive: as $x
ightarrow-\infty$, $y
ightarrow-\infty$; as $x
ightarrow\infty$, $y
ightarrow\infty$
    - If the leading coefficient is negative: as $x
ightarrow-\infty$, $y
ightarrow\infty$; as $x
ightarrow\infty$, $y
ightarrow-\infty$

If there is a relative minimum, and we look at the options:

Option C: As $x
ightarrow-\infty,y
ightarrow\infty$ and as $x
ightarrow\infty,y
ightarrow-\infty$ (odd - degree, negative leading coefficient) can have a relative minimum (and a relative maximum). Option D: As $x
ightarrow-\infty,y
ightarrow\infty$ and as $x
ightarrow\infty,y
ightarrow\infty$ (even - degree, positive leading coefficient) has a minimum. But from the first part, we have "No Relative minima" is incorrect (the first part has "No Relative minima" selected, but maybe that's a mistake). Wait, the first part has "No Relative minima" selected, but then $x = 0,y=-5$. That's a contradiction. But assuming that the correct end - behavior is for a function where as $x
ightarrow-\infty$, $y
ightarrow\infty$ and as $x
ightarrow\infty$, $y
ightarrow-\infty$ (Option C) or maybe the correct answer is C. Wait, no, let's think again. If the function has a relative minimum at $x = 0,y=-5$, and we consider the end - behavior. If the function is a cubic with positive leading coefficient, as $x
ightarrow-\infty$, $y
ightarrow-\infty$; as $x
ightarrow\infty$, $y
ightarrow\infty$ (Option B), and it can have a relative minimum and a relative maximum. But the first part says "No Relative maxima" (selected as "none") and "No Relative minima" (but then $x = 0,y=-5$). This is a bit confusing, but among the options, the most probable end - behavior for a function with a relative minimum (assuming the "No Relative minima" is a mistake) and the options, if we consider the graph (maybe a cubic with negative leading coefficient, which has a relative max and min, but the "No Relative maxima" is wrong, but among the options, Option C: As $x
ightarrow-\infty,y
ightarrow\infty$ and as $x
ightarrow\infty,y
ightarrow-\infty$ is the behavior of an odd - degree function with negative leading coefficient, which can have a relative minimum (and a relative maximum). So the correct option is C.

# Answer:
C. As $x \to -\infty$, $y \to \infty$ and as $x \to \infty$, $y \to -\infty$

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