Tom Started An Entertainment Company. The Net Value Of The Company (in Thousands Of Dollars) Months After Its Creation Is Modeled By $v(t)=4t^2-24t-28$. Tom Wants To Know When His Company Will Be At Its Lowest Net Value. How Many Months After

Introduction

Tom started an entertainment company, and the net value of the company is modeled by the quadratic function $v(t)=4t^2-24t-28$, where $t$ represents the number of months after its creation. Tom wants to know when his company will be at its lowest net value. In this article, we will explore how to find the minimum value of the quadratic function and determine when the company will reach its lowest net value.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. In this case, the quadratic function is $v(t)=4t^2-24t-28$, where $a=4$, $b=-24$, and $c=-28$.

Finding the Minimum Value of a Quadratic Function

To find the minimum value of a quadratic function, we need to find the vertex of the parabola. The vertex is the point on the parabola where the function changes from decreasing to increasing or vice versa. The x-coordinate of the vertex can be found using the formula $x=-\frac{b}{2a}$. In this case, $a=4$ and $b=-24$, so the x-coordinate of the vertex is $x=-\frac{-24}{2(4)}=\frac{24}{8}=3$.

Finding the Minimum Value of the Function

Now that we have found the x-coordinate of the vertex, we can find the minimum value of the function by plugging this value back into the function. The minimum value of the function is $v(3)=4(3)^2-24(3)-28=36-72-28=-64$. Therefore, the minimum value of the function is $-64$, and this occurs when $t=3$.

Interpreting the Results

The results indicate that the company will be at its lowest net value when $t=3$ months after its creation. This means that the company will reach its lowest net value in 3 months. It's essential to note that this is a mathematical model, and the actual performance of the company may vary.

Conclusion

In conclusion, we have found the minimum value of the quadratic function $v(t)=4t^2-24t-28$ and determined when the company will reach its lowest net value. The minimum value of the function is $-64$, and this occurs when $t=3$ months after its creation. This information can be useful for Tom to make informed decisions about his company's operations and financial planning.

Additional Considerations

While this mathematical model provides valuable insights, it's essential to consider other factors that may affect the company's performance. These factors may include market trends, competition, and economic conditions. A more comprehensive analysis of the company's financial performance would require considering these factors and other relevant data.

Mathematical Derivations

For those interested in the mathematical derivations, we can derive the formula for the x-coordinate of the vertex using calculus. The derivative of the function is $v'(t)=8t-24$. Setting the derivative equal to zero, we get $8t-24=0$, which gives us $t=3$. This confirms the result obtained using the formula $x=-\frac{b}{2a}$.

Real-World Applications

The concept of finding the minimum value of a quadratic function has numerous real-world applications. For example, in economics, the minimum value of a quadratic function can represent the minimum cost or maximum profit of a business. In engineering, the minimum value of a quadratic function can represent the minimum stress or maximum strain on a structure. In finance, the minimum value of a quadratic function can represent the minimum risk or maximum return on an investment.

Conclusion

Q: What is the minimum value of the quadratic function $v(t)=4t^2-24t-28$?

A: The minimum value of the quadratic function is $-64$. This occurs when $t=3$ months after its creation.

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

A: To find the minimum value of a quadratic function, you need to find the vertex of the parabola. The x-coordinate of the vertex can be found using the formula $x=-\frac{b}{2a}$. Once you have the x-coordinate, you can plug this value back into the function to find the minimum value.

Q: What is the significance of the x-coordinate of the vertex?

A: The x-coordinate of the vertex represents the point on the parabola where the function changes from decreasing to increasing or vice versa. This is the point where the function reaches its minimum or maximum value.

Q: Can I use calculus to find the minimum value of a quadratic function?

A: Yes, you can use calculus to find the minimum value of a quadratic function. The derivative of the function is $v'(t)=8t-24$. Setting the derivative equal to zero, you can find the x-coordinate of the vertex, which is $t=3$ in this case.

Q: How does the minimum value of a quadratic function relate to real-world applications?

A: The minimum value of a quadratic function has numerous real-world applications. For example, in economics, the minimum value of a quadratic function can represent the minimum cost or maximum profit of a business. In engineering, the minimum value of a quadratic function can represent the minimum stress or maximum strain on a structure. In finance, the minimum value of a quadratic function can represent the minimum risk or maximum return on an investment.

Q: Can I use the quadratic function $v(t)=4t^2-24t-28$ to model other real-world scenarios?

A: Yes, you can use the quadratic function $v(t)=4t^2-24t-28$ to model other real-world scenarios. For example, you can use this function to model the growth or decline of a population, the spread of a disease, or the movement of an object under the influence of gravity.

Q: How do I interpret the results of a quadratic function?

A: To interpret the results of a quadratic function, you need to consider the context of the problem. For example, if you are using the function to model the growth of a population, you need to consider the rate of growth and the maximum population size. If you are using the function to model the spread of a disease, you need to consider the rate of spread and the maximum number of infected individuals.

Q: Can I use technology to graph and analyze quadratic functions?

A: Yes, you can use technology to graph and analyze quadratic functions. Graphing calculators and computer software can be used to visualize the graph of a quadratic function and analyze its properties.

Q: How do I choose the right quadratic function to model a real-world scenario?

A: To choose the right quadratic function to model a real-world scenario, you need to consider the characteristics of the scenario. For example, if you are modeling the growth of a population, you need to consider the rate of growth and the maximum population size. If you are modeling the spread of a disease, you need to consider the rate of spread and the maximum number of infected individuals.

Conclusion

In conclusion, we have explored the quadratic function $v(t)=4t^2-24t-28$ and its applications in real-world scenarios. We have also answered common questions about quadratic functions and their properties. By understanding the characteristics of quadratic functions, you can use them to model and analyze a wide range of real-world scenarios.