Question
Use the graph to answer the question. When appropriate, include why the answer you choose is correct and why the other choices are not correct. Which of the following describes the end behavior of this function? "x" to negative infinity , "y" to negative infinity"x" to infinity , "y" to infinity "x" to negative infinity , "y" to infinity"x" to infinity , "y" to infinity "x" to negative infinity , "y" to infinity"x" to infinity , "y" to negative infinity "x" to negative infinity , "y" to negative infinity"x" to infinity , "y" to negative infinity
Answers
GPT-4o mini
To analyze the end behavior of a function, we focus on how the values of \( y \) behave as \( x \) approaches both positive and negative infinity. Since I cannot see the actual graph, I will provide guidance on how to deduce the correct choice based on common end behaviors of functions.
1. **"x" to negative infinity, "y" to negative infinity**: This describes a function that decreases without bound as \( x \) approaches negative infinity. This is typical for some polynomial functions like a downward-opening parabola.
2. **"x" to infinity, "y" to infinity**: This means that the function increases without bound as \( x \) approaches positive infinity, which might also appear for polynomial functions, especially those of even degree with a positive leading coefficient.
3. **"x" to negative infinity, "y" to infinity**: This indicates that as \( x \) goes to negative infinity, \( y \) increases without bound, often seen in functions that have odd degrees, like certain cubic functions.
4. **"x" to infinity, "y" to infinity**: Similar to point 2, this reflects the function behaving positively as \( x \) approaches positive infinity.
5. **"x" to negative infinity, "y" to infinity; "x" to infinity, "y" to negative infinity**: This describes a function that behaves oppositely at the two extremes of \( x \). A typical example could be a rational function or an odd-degree polynomial where one end rises while the other falls.
6. **"x" to negative infinity, "y" to negative infinity; "x" to infinity, "y" to negative infinity**: This indicates that the function continues to decrease without bound as \( x \) increases on both extremes, often observed in functions going downwards.
To choose the correct description, observe the graph's end behavior visually. Look for arrows at each end of the graph: if they point downwards, select an option indicating \( y \to -\infty \); if they point upwards, choose one indicating \( y \to +\infty \).
### Example Analysis:
- If the graph shows that as \( x \) goes to negative infinity, the values of \( y \) go downwards, and as \( x \) increases, the values of \( y \) also go downwards, you would choose: **"x" to negative infinity, "y" to negative infinity; "x" to infinity, "y" to negative infinity**.
Choosing the most accurate description of the function involves carefully interpreting how \( y \) behaves at both extremes of the \( x \)-axis based on visual input from the graph.
1. **"x" to negative infinity, "y" to negative infinity**: This describes a function that decreases without bound as \( x \) approaches negative infinity. This is typical for some polynomial functions like a downward-opening parabola.
2. **"x" to infinity, "y" to infinity**: This means that the function increases without bound as \( x \) approaches positive infinity, which might also appear for polynomial functions, especially those of even degree with a positive leading coefficient.
3. **"x" to negative infinity, "y" to infinity**: This indicates that as \( x \) goes to negative infinity, \( y \) increases without bound, often seen in functions that have odd degrees, like certain cubic functions.
4. **"x" to infinity, "y" to infinity**: Similar to point 2, this reflects the function behaving positively as \( x \) approaches positive infinity.
5. **"x" to negative infinity, "y" to infinity; "x" to infinity, "y" to negative infinity**: This describes a function that behaves oppositely at the two extremes of \( x \). A typical example could be a rational function or an odd-degree polynomial where one end rises while the other falls.
6. **"x" to negative infinity, "y" to negative infinity; "x" to infinity, "y" to negative infinity**: This indicates that the function continues to decrease without bound as \( x \) increases on both extremes, often observed in functions going downwards.
To choose the correct description, observe the graph's end behavior visually. Look for arrows at each end of the graph: if they point downwards, select an option indicating \( y \to -\infty \); if they point upwards, choose one indicating \( y \to +\infty \).
### Example Analysis:
- If the graph shows that as \( x \) goes to negative infinity, the values of \( y \) go downwards, and as \( x \) increases, the values of \( y \) also go downwards, you would choose: **"x" to negative infinity, "y" to negative infinity; "x" to infinity, "y" to negative infinity**.
Choosing the most accurate description of the function involves carefully interpreting how \( y \) behaves at both extremes of the \( x \)-axis based on visual input from the graph.