Answer :
To address the given problem involving the profit function for a shopping mall, [tex]\( S(x) = -x^2 + 36x - 288 \)[/tex], we will examine and graph several key features of the function.
### Key Feature Analysis:
1. Interval(s) where the function is decreasing:
The profit function [tex]\( S(x) \)[/tex] is a downward-opening parabola. We identify the vertex first to determine where the function changes from increasing to decreasing. With the vertex at [tex]\( x = 18 \)[/tex], the function is decreasing for [tex]\( x > 18 \)[/tex].
Decreasing interval: [tex]\((18, \infty)\)[/tex].
2. Interval(s) where the function is positive:
To find where [tex]\( S(x) \)[/tex] is positive, we solve [tex]\( -x^2 + 36x - 288 > 0 \)[/tex]. The roots of the equation are [tex]\( x = 12 \)[/tex] and [tex]\( x = 24 \)[/tex]. Therefore, [tex]\( S(x) \)[/tex] is positive between these two roots.
Positive interval: [tex]\((12, 24)\)[/tex].
3. Interval(s) where the function is negative:
Similarly, we find where [tex]\( S(x) \)[/tex] is negative. For [tex]\( x \)[/tex] values outside of the positive interval, [tex]\( S(x) \)[/tex] will be negative.
Negative intervals: [tex]\((-\infty, 12) \cup (24, \infty)\)[/tex].
4. Vertex:
The vertex, being the maximum point of the parabola, occurs at [tex]\( x = 18 \)[/tex]. Plugging [tex]\( x = 18 \)[/tex] back into the function [tex]\( S(x) \)[/tex] gives [tex]\( S(18) = 36 \)[/tex].
Vertex: [tex]\((18, 36)\)[/tex].
5. Axis of symmetry:
The axis of symmetry for the parabola is the vertical line that passes through the vertex. Hence, it occurs at [tex]\( x = 18 \)[/tex].
Axis of symmetry: [tex]\( x = 18 \)[/tex].
### Graphing [tex]\( S(x) \)[/tex] with Key Features:
- The graph of [tex]\( S(x) = -x^2 + 36x - 288 \)[/tex] is a parabola opening downwards.
- The vertex [tex]\((18, 36)\)[/tex] is the highest point on the graph.
- The axis of symmetry is the vertical line [tex]\( x = 18 \)[/tex].
- The function is positive for intervals [tex]\((12, 24)\)[/tex].
- The function is negative for intervals [tex]\((-\infty, 12)\)[/tex] and [tex]\((24, \infty)\)[/tex].
- The function is decreasing on the interval [tex]\((18, \infty)\)[/tex].
Below is a detailed depiction of the key features:
1. The parabola (profit function) is drawn with a peak at [tex]\((18, 36)\)[/tex].
2. The axis of symmetry is shown as a dashed vertical line at [tex]\( x = 18 \)[/tex].
3. Positive intervals [tex]\((12,24)\)[/tex] where the function is above the x-axis are highlighted.
4. Negative intervals [tex]\((-\infty,12)\)[/tex] and [tex]\((24,\infty)\)[/tex] where the function is below the x-axis are indicated.
5. The interval where the function is decreasing, [tex]\( (18, \infty) \)[/tex], is marked on the graph.
By examining these features, you can better understand the behavior of the profit function and make more informed decisions about the number of retail stores to include in the shopping mall to optimize profitability.
### Key Feature Analysis:
1. Interval(s) where the function is decreasing:
The profit function [tex]\( S(x) \)[/tex] is a downward-opening parabola. We identify the vertex first to determine where the function changes from increasing to decreasing. With the vertex at [tex]\( x = 18 \)[/tex], the function is decreasing for [tex]\( x > 18 \)[/tex].
Decreasing interval: [tex]\((18, \infty)\)[/tex].
2. Interval(s) where the function is positive:
To find where [tex]\( S(x) \)[/tex] is positive, we solve [tex]\( -x^2 + 36x - 288 > 0 \)[/tex]. The roots of the equation are [tex]\( x = 12 \)[/tex] and [tex]\( x = 24 \)[/tex]. Therefore, [tex]\( S(x) \)[/tex] is positive between these two roots.
Positive interval: [tex]\((12, 24)\)[/tex].
3. Interval(s) where the function is negative:
Similarly, we find where [tex]\( S(x) \)[/tex] is negative. For [tex]\( x \)[/tex] values outside of the positive interval, [tex]\( S(x) \)[/tex] will be negative.
Negative intervals: [tex]\((-\infty, 12) \cup (24, \infty)\)[/tex].
4. Vertex:
The vertex, being the maximum point of the parabola, occurs at [tex]\( x = 18 \)[/tex]. Plugging [tex]\( x = 18 \)[/tex] back into the function [tex]\( S(x) \)[/tex] gives [tex]\( S(18) = 36 \)[/tex].
Vertex: [tex]\((18, 36)\)[/tex].
5. Axis of symmetry:
The axis of symmetry for the parabola is the vertical line that passes through the vertex. Hence, it occurs at [tex]\( x = 18 \)[/tex].
Axis of symmetry: [tex]\( x = 18 \)[/tex].
### Graphing [tex]\( S(x) \)[/tex] with Key Features:
- The graph of [tex]\( S(x) = -x^2 + 36x - 288 \)[/tex] is a parabola opening downwards.
- The vertex [tex]\((18, 36)\)[/tex] is the highest point on the graph.
- The axis of symmetry is the vertical line [tex]\( x = 18 \)[/tex].
- The function is positive for intervals [tex]\((12, 24)\)[/tex].
- The function is negative for intervals [tex]\((-\infty, 12)\)[/tex] and [tex]\((24, \infty)\)[/tex].
- The function is decreasing on the interval [tex]\((18, \infty)\)[/tex].
Below is a detailed depiction of the key features:
1. The parabola (profit function) is drawn with a peak at [tex]\((18, 36)\)[/tex].
2. The axis of symmetry is shown as a dashed vertical line at [tex]\( x = 18 \)[/tex].
3. Positive intervals [tex]\((12,24)\)[/tex] where the function is above the x-axis are highlighted.
4. Negative intervals [tex]\((-\infty,12)\)[/tex] and [tex]\((24,\infty)\)[/tex] where the function is below the x-axis are indicated.
5. The interval where the function is decreasing, [tex]\( (18, \infty) \)[/tex], is marked on the graph.
By examining these features, you can better understand the behavior of the profit function and make more informed decisions about the number of retail stores to include in the shopping mall to optimize profitability.
