Answer :
Let's analyze the function [tex]\( f(x) = -|x + 3| + 5 \)[/tex] step-by-step to determine whether Lared's characteristics are correct.
### 1. Vertex
The function [tex]\( f(x) = -|x + 3| + 5 \)[/tex] can be written in the form of a vertex form [tex]\( f(x) = -|x - h| + k \)[/tex].
- Here, we see that the [tex]\( x \)[/tex]-coordinate of the vertex [tex]\( h \)[/tex] is such that [tex]\( x + 3 = 0 \)[/tex]. Solving this, [tex]\( x = -3 \)[/tex].
- The [tex]\( y \)[/tex]-coordinate of the vertex is [tex]\( k \)[/tex], which is given as 5.
Therefore, the vertex of [tex]\( f(x) = -|x + 3| + 5 \)[/tex] is [tex]\( (-3, 5) \)[/tex].
### 2. Minimum/Maximum
The function [tex]\( f(x) = -|x + 3| + 5 \)[/tex] represents a downward opening absolute value function.
- For downward opening absolute value functions like [tex]\( -|x| \)[/tex], the highest point (maximum value) is at the vertex.
- Hence, the maximum value of the function is the [tex]\( y \)[/tex]-coordinate of the vertex, which is 5.
- The function does not have a minimum value because it extends infinitely downwards.
Therefore, the maximum value is correctly identified as 5.
### 3. Interval of Decrease
The function [tex]\( f(x) = -|x + 3| + 5 \)[/tex] has an absolute value form that decreases from its vertex.
- The vertex is at [tex]\( x = -3 \)[/tex].
- The function decreases as [tex]\( x \)[/tex] moves away from the vertex on both sides.
- Specifically, it decreases on the intervals [tex]\( (-\infty, -3) \)[/tex] and [tex]\( (-3, + \infty) \)[/tex].
As per the student's response that the function decreases for [tex]\( x < -3 \)[/tex], this is only partially correct. In reality, it decreases both when [tex]\( x < -3 \)[/tex] and [tex]\( x > -3 \)[/tex].
### 4. Domain
The domain of an absolute value function is all real numbers because absolute values are defined for all real numbers.
Thus, the domain is correctly identified as all real numbers [tex]\( (-\infty, +\infty) \)[/tex].
### Corrections
- The vertex is correctly identified as [tex]\((-3, 5)\)[/tex].
- The maximum is correctly identified as 5.
- The interval of decrease should include both intervals: for [tex]\( x < -3 \)[/tex] and for [tex]\( x > -3 \)[/tex].
- The domain is correctly identified as all real numbers.
### Conclusion
Despite the suggestion of corrections, the provided analysis concludes that Lared's initial analysis is accurate with regards to the vertex, the maximum value, the domain, but is partially correct about the interval of decrease. Hence, the interval of decrease should be expanded to cover both [tex]\( x < -3 \)[/tex] and [tex]\( x > -3 \)[/tex].
### 1. Vertex
The function [tex]\( f(x) = -|x + 3| + 5 \)[/tex] can be written in the form of a vertex form [tex]\( f(x) = -|x - h| + k \)[/tex].
- Here, we see that the [tex]\( x \)[/tex]-coordinate of the vertex [tex]\( h \)[/tex] is such that [tex]\( x + 3 = 0 \)[/tex]. Solving this, [tex]\( x = -3 \)[/tex].
- The [tex]\( y \)[/tex]-coordinate of the vertex is [tex]\( k \)[/tex], which is given as 5.
Therefore, the vertex of [tex]\( f(x) = -|x + 3| + 5 \)[/tex] is [tex]\( (-3, 5) \)[/tex].
### 2. Minimum/Maximum
The function [tex]\( f(x) = -|x + 3| + 5 \)[/tex] represents a downward opening absolute value function.
- For downward opening absolute value functions like [tex]\( -|x| \)[/tex], the highest point (maximum value) is at the vertex.
- Hence, the maximum value of the function is the [tex]\( y \)[/tex]-coordinate of the vertex, which is 5.
- The function does not have a minimum value because it extends infinitely downwards.
Therefore, the maximum value is correctly identified as 5.
### 3. Interval of Decrease
The function [tex]\( f(x) = -|x + 3| + 5 \)[/tex] has an absolute value form that decreases from its vertex.
- The vertex is at [tex]\( x = -3 \)[/tex].
- The function decreases as [tex]\( x \)[/tex] moves away from the vertex on both sides.
- Specifically, it decreases on the intervals [tex]\( (-\infty, -3) \)[/tex] and [tex]\( (-3, + \infty) \)[/tex].
As per the student's response that the function decreases for [tex]\( x < -3 \)[/tex], this is only partially correct. In reality, it decreases both when [tex]\( x < -3 \)[/tex] and [tex]\( x > -3 \)[/tex].
### 4. Domain
The domain of an absolute value function is all real numbers because absolute values are defined for all real numbers.
Thus, the domain is correctly identified as all real numbers [tex]\( (-\infty, +\infty) \)[/tex].
### Corrections
- The vertex is correctly identified as [tex]\((-3, 5)\)[/tex].
- The maximum is correctly identified as 5.
- The interval of decrease should include both intervals: for [tex]\( x < -3 \)[/tex] and for [tex]\( x > -3 \)[/tex].
- The domain is correctly identified as all real numbers.
### Conclusion
Despite the suggestion of corrections, the provided analysis concludes that Lared's initial analysis is accurate with regards to the vertex, the maximum value, the domain, but is partially correct about the interval of decrease. Hence, the interval of decrease should be expanded to cover both [tex]\( x < -3 \)[/tex] and [tex]\( x > -3 \)[/tex].
