Answer :
To determine which graph would be associated with the given quadratic function that models the number of customers visiting a store each hour, we need to analyze the function step-by-step. The function is:
[tex]\[ n(t) = -2.82 t^2 + 25.74 t + 60.87 \][/tex]
Let's examine the key characteristics of this quadratic function:
1. Form of the Function:
The quadratic function is in the standard form [tex]\( n(t) = at^2 + bt + c \)[/tex], where:
- [tex]\( a = -2.82 \)[/tex]
- [tex]\( b = 25.74 \)[/tex]
- [tex]\( c = 60.87 \)[/tex]
2. Direction of the Parabola:
Since the coefficient [tex]\( a \)[/tex] (which is -2.82) is negative, the parabola opens downwards. This means the number of customers increases to a maximum point and then decreases.
3. Vertex of the Parabola:
The vertex of the quadratic function [tex]\( n(t) = at^2 + bt + c \)[/tex] gives us the point where the maximum number of customers occurs. The vertex formula for a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is given by:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Substituting [tex]\( a = -2.82 \)[/tex] and [tex]\( b = 25.74 \)[/tex]:
[tex]\[ t = -\frac{25.74}{2 \times -2.82} \][/tex]
[tex]\[ t = \frac{25.74}{5.64} \][/tex]
[tex]\[ t \approx 4.56 \][/tex]
This means the maximum number of customers occurs approximately at [tex]\( t = 4.56 \)[/tex] hours past the start of the observation period.
4. Value of the Maximum Number of Customers:
To find the maximum number of customers, substitute [tex]\( t = 4.56 \)[/tex] back into the function [tex]\( n(t) \)[/tex]:
[tex]\[ n(4.56) = -2.82 (4.56)^2 + 25.74 (4.56) + 60.87 \][/tex]
Calculate [tex]\( (4.56)^2 \)[/tex]:
[tex]\[ (4.56)^2 = 20.7936 \][/tex]
Now,
[tex]\[ n(4.56) = -2.82 \times 20.7936 + 25.74 \times 4.56 + 60.87 \][/tex]
[tex]\[ n(4.56) = -58.6386 + 117.3744 + 60.87 \][/tex]
[tex]\[ n(4.56) \approx 119.61 \][/tex]
Thus, the maximum number of customers is approximately 119.
5. General Behavior:
- At [tex]\( t = 0 \)[/tex] (the start of the observation period), the number of customers is:
[tex]\[ n(0) = 60.87 \][/tex]
- As [tex]\( t \)[/tex] increases from 0, the number of customers initially rises, reaches a peak, and then falls as [tex]\( t \)[/tex] continues to increase beyond the vertex.
From these steps, we can summarize the graph's characteristics:
- It is a downward-opening parabola (i.e., an inverted U-shape).
- It has a maximum point (vertex) at [tex]\( t \approx 4.56 \)[/tex] with the maximum number of customers being around 119.
- At [tex]\( t = 0 \)[/tex], the graph starts at [tex]\( n(t) = 60.87 \)[/tex].
Thus, the correct graph would be a parabola that starts at around 60.87 on the y-axis, peaks at [tex]\( t \approx 4.56 \)[/tex] with a maximum value of around 119, and then declines as [tex]\( t \)[/tex] increases. The graph should reflect an initial increase, a peak, and then a decrease, matching the described behavior of the store's customer visits over the hours.
[tex]\[ n(t) = -2.82 t^2 + 25.74 t + 60.87 \][/tex]
Let's examine the key characteristics of this quadratic function:
1. Form of the Function:
The quadratic function is in the standard form [tex]\( n(t) = at^2 + bt + c \)[/tex], where:
- [tex]\( a = -2.82 \)[/tex]
- [tex]\( b = 25.74 \)[/tex]
- [tex]\( c = 60.87 \)[/tex]
2. Direction of the Parabola:
Since the coefficient [tex]\( a \)[/tex] (which is -2.82) is negative, the parabola opens downwards. This means the number of customers increases to a maximum point and then decreases.
3. Vertex of the Parabola:
The vertex of the quadratic function [tex]\( n(t) = at^2 + bt + c \)[/tex] gives us the point where the maximum number of customers occurs. The vertex formula for a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is given by:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Substituting [tex]\( a = -2.82 \)[/tex] and [tex]\( b = 25.74 \)[/tex]:
[tex]\[ t = -\frac{25.74}{2 \times -2.82} \][/tex]
[tex]\[ t = \frac{25.74}{5.64} \][/tex]
[tex]\[ t \approx 4.56 \][/tex]
This means the maximum number of customers occurs approximately at [tex]\( t = 4.56 \)[/tex] hours past the start of the observation period.
4. Value of the Maximum Number of Customers:
To find the maximum number of customers, substitute [tex]\( t = 4.56 \)[/tex] back into the function [tex]\( n(t) \)[/tex]:
[tex]\[ n(4.56) = -2.82 (4.56)^2 + 25.74 (4.56) + 60.87 \][/tex]
Calculate [tex]\( (4.56)^2 \)[/tex]:
[tex]\[ (4.56)^2 = 20.7936 \][/tex]
Now,
[tex]\[ n(4.56) = -2.82 \times 20.7936 + 25.74 \times 4.56 + 60.87 \][/tex]
[tex]\[ n(4.56) = -58.6386 + 117.3744 + 60.87 \][/tex]
[tex]\[ n(4.56) \approx 119.61 \][/tex]
Thus, the maximum number of customers is approximately 119.
5. General Behavior:
- At [tex]\( t = 0 \)[/tex] (the start of the observation period), the number of customers is:
[tex]\[ n(0) = 60.87 \][/tex]
- As [tex]\( t \)[/tex] increases from 0, the number of customers initially rises, reaches a peak, and then falls as [tex]\( t \)[/tex] continues to increase beyond the vertex.
From these steps, we can summarize the graph's characteristics:
- It is a downward-opening parabola (i.e., an inverted U-shape).
- It has a maximum point (vertex) at [tex]\( t \approx 4.56 \)[/tex] with the maximum number of customers being around 119.
- At [tex]\( t = 0 \)[/tex], the graph starts at [tex]\( n(t) = 60.87 \)[/tex].
Thus, the correct graph would be a parabola that starts at around 60.87 on the y-axis, peaks at [tex]\( t \approx 4.56 \)[/tex] with a maximum value of around 119, and then declines as [tex]\( t \)[/tex] increases. The graph should reflect an initial increase, a peak, and then a decrease, matching the described behavior of the store's customer visits over the hours.
