Asked by monday
Select the correct response. The graph of the function y=−2(x+4)−−−−−−√3 is increasing/decreasing explain your answer
Answers
Answered by
GPT-4o mini
To analyze whether the function \( y = -2\sqrt{x + 4} - 3 \) is increasing or decreasing, we can examine the structure of the function.
1. **Function Form**: The function contains a square root, \( \sqrt{x + 4} \), which is a monotonically increasing function over its domain (where \( x + 4 \geq 0 \) or \( x \geq -4 \)).
2. **Coefficient of \(-2\)**: The square root is multiplied by \(-2\). This reflection across the x-axis indicates that the output of the function decreases as \( \sqrt{x + 4} \) increases.
3. **Vertical Shift**: The entire function is then shifted down by 3 units due to the \(-3\) at the end. This does not affect the increasing or decreasing nature of the function.
### Conclusion:
Since we have a negative coefficient in front of the square root, the function \( y = -2\sqrt{x + 4} - 3 \) is decreasing. As \( x \) increases (within its domain), \( y \) decreases.
Thus, the graph of the function is **decreasing**.
**Explanation**: Because the square root function is increasing and is multiplied by -2, it reverses the behavior to decreasing.
1. **Function Form**: The function contains a square root, \( \sqrt{x + 4} \), which is a monotonically increasing function over its domain (where \( x + 4 \geq 0 \) or \( x \geq -4 \)).
2. **Coefficient of \(-2\)**: The square root is multiplied by \(-2\). This reflection across the x-axis indicates that the output of the function decreases as \( \sqrt{x + 4} \) increases.
3. **Vertical Shift**: The entire function is then shifted down by 3 units due to the \(-3\) at the end. This does not affect the increasing or decreasing nature of the function.
### Conclusion:
Since we have a negative coefficient in front of the square root, the function \( y = -2\sqrt{x + 4} - 3 \) is decreasing. As \( x \) increases (within its domain), \( y \) decreases.
Thus, the graph of the function is **decreasing**.
**Explanation**: Because the square root function is increasing and is multiplied by -2, it reverses the behavior to decreasing.