Select The Correct Answer.Martha Manages A Home Improvement Store And Uses This Function To Model The Number Of Customers In The Afternoon: N ( T ) = − 2.82 T 2 + 25.74 T + 60.87 N(t) = -2.82t^2 + 25.74t + 60.87 N ( T ) = − 2.82 T 2 + 25.74 T + 60.87 Which Graph Would Most Likely Be Associated With This Model? A.

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Introduction

In the field of home improvement store management, understanding the behavior of customer traffic is crucial for making informed decisions. One way to model this behavior is by using quadratic functions, which can help predict the number of customers at a given time. In this article, we will explore how to select the correct graph associated with a quadratic function used to model the number of customers in the afternoon.

The Quadratic Function

The quadratic function used to model the number of customers in the afternoon is given by:

n(t)=−2.82t2+25.74t+60.87n(t) = -2.82t^2 + 25.74t + 60.87

where n(t)n(t) represents the number of customers at time tt.

Graphing Quadratic Functions

To graph a quadratic function, we need to identify its key features, such as the vertex, axis of symmetry, and direction of opening. The general form of a quadratic function is:

f(x)=ax2+bx+cf(x) = ax^2 + bx + c

where aa, bb, and cc are constants.

Vertex Form

The vertex form of a quadratic function is:

f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k

where (h,k)(h, k) is the vertex of the parabola.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. It can be found using the formula:

x=−b2ax = -\frac{b}{2a}

Direction of Opening

The direction of opening of a parabola is determined by the sign of the coefficient aa. If aa is positive, the parabola opens upward. If aa is negative, the parabola opens downward.

Graphing the Quadratic Function

To graph the quadratic function n(t)=−2.82t2+25.74t+60.87n(t) = -2.82t^2 + 25.74t + 60.87, we need to identify its key features.

  • Vertex: To find the vertex, we need to complete the square or use the formula t=−b2at = -\frac{b}{2a}. In this case, a=−2.82a = -2.82 and b=25.74b = 25.74. Plugging these values into the formula, we get: t=−25.742(−2.82)=4.57t = -\frac{25.74}{2(-2.82)} = 4.57. Therefore, the vertex is at (4.57,n(4.57))(4.57, n(4.57)).
  • Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex. In this case, the axis of symmetry is t=4.57t = 4.57.
  • Direction of Opening: Since a=−2.82a = -2.82 is negative, the parabola opens downward.

Selecting the Correct Graph

Based on the key features of the quadratic function, we can select the correct graph.

  • Graph A: This graph has a vertex at (4.57,n(4.57))(4.57, n(4.57)), an axis of symmetry at t=4.57t = 4.57, and opens downward. Therefore, this graph is most likely associated with the quadratic function n(t)=−2.82t2+25.74t+60.87n(t) = -2.82t^2 + 25.74t + 60.87.

Conclusion

In conclusion, understanding the graph of a quadratic function is crucial in home improvement store management. By identifying the key features of the quadratic function, such as the vertex, axis of symmetry, and direction of opening, we can select the correct graph associated with the model. In this article, we explored how to graph a quadratic function and select the correct graph associated with a model used to predict the number of customers in the afternoon.

Key Takeaways

  • The quadratic function n(t)=−2.82t2+25.74t+60.87n(t) = -2.82t^2 + 25.74t + 60.87 models the number of customers in the afternoon.
  • The graph of the quadratic function has a vertex at (4.57,n(4.57))(4.57, n(4.57)), an axis of symmetry at t=4.57t = 4.57, and opens downward.
  • Graph A is the most likely associated with the quadratic function n(t)=−2.82t2+25.74t+60.87n(t) = -2.82t^2 + 25.74t + 60.87.

References

Additional Resources

Introduction

In our previous article, we explored how to graph a quadratic function and select the correct graph associated with a model used to predict the number of customers in the afternoon. In this article, we will answer some frequently asked questions about quadratic functions and their applications in home improvement store management.

Q&A

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means it has a highest power of two. It is typically written in the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point at which the function reaches its maximum or minimum value. It is typically denoted as (h,k)(h, k), where hh is the x-coordinate and kk is the y-coordinate.

Q: How do I find the vertex of a quadratic function?

A: To find the vertex of a quadratic function, you can use the formula t=−b2at = -\frac{b}{2a}, where aa and bb are the coefficients of the quadratic function.

Q: What is the axis of symmetry?

A: The axis of symmetry is a vertical line that passes through the vertex of a quadratic function. It is typically denoted as x=hx = h, where hh is the x-coordinate of the vertex.

Q: How do I determine the direction of opening of a quadratic function?

A: To determine the direction of opening of a quadratic function, you can look at the sign of the coefficient aa. If aa is positive, the parabola opens upward. If aa is negative, the parabola opens downward.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use the following steps:

  1. Find the vertex of the function.
  2. Find the axis of symmetry.
  3. Determine the direction of opening.
  4. Plot the points on a coordinate plane.

Q: What is the significance of the vertex in home improvement store management?

A: The vertex of a quadratic function represents the maximum or minimum number of customers at a given time. In home improvement store management, this information can be used to determine the optimal time to restock shelves or schedule staff.

Q: How can I use quadratic functions in home improvement store management?

A: Quadratic functions can be used to model the number of customers at a given time, which can help you make informed decisions about staffing, inventory, and marketing.

Q: What are some common applications of quadratic functions in home improvement store management?

A: Some common applications of quadratic functions in home improvement store management include:

  • Modeling customer traffic
  • Determining optimal staffing levels
  • Scheduling restocking and inventory management
  • Analyzing sales data

Conclusion

In conclusion, quadratic functions are a powerful tool in home improvement store management. By understanding the graph of a quadratic function, you can make informed decisions about staffing, inventory, and marketing. We hope this Q&A article has provided you with a better understanding of quadratic functions and their applications in home improvement store management.

Key Takeaways

  • Quadratic functions are a polynomial function of degree two.
  • The vertex of a quadratic function represents the maximum or minimum value.
  • The axis of symmetry is a vertical line that passes through the vertex.
  • The direction of opening is determined by the sign of the coefficient aa.
  • Quadratic functions can be used to model customer traffic and determine optimal staffing levels.

References

Additional Resources