Substitute 10 For $b$ And $-\frac{1}{2}$ For $a$.$-\frac{b}{2a} = -\frac{10}{2\left(-\frac{1}{2}\right)} = -\frac{10}{-1} = 10$The $y$-coordinate Of The Vertex Gives The Greatest Possible Weekly Profit For

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In mathematics, the vertex form of a quadratic function is a powerful tool for analyzing and solving quadratic equations. It provides valuable insights into the behavior of the function, including its maximum or minimum value, which is essential in various real-world applications, such as finance, engineering, and economics. In this article, we will delve into the concept of the vertex form of a quadratic function, explore its significance, and provide a step-by-step guide on how to substitute values into the equation.

What is the Vertex Form of a Quadratic Function?

The vertex form of a quadratic function is a mathematical representation of a quadratic equation in the form of:

f(x) = a(x - h)^2 + k

where:

  • f(x) is the function
  • a is the coefficient of the squared term
  • h is the x-coordinate of the vertex
  • k is the y-coordinate of the vertex

The vertex form is particularly useful in identifying the maximum or minimum value of a quadratic function, which is essential in various applications, such as finance, where the maximum profit or minimum cost is a critical factor in decision-making.

Substituting Values into the Equation

In the given problem, we are asked to substitute 10 for b and -1/2 for a in the equation -b/2a. To do this, we need to follow the order of operations (PEMDAS):

  1. Substitute 10 for b: -b/2a becomes -10/2a
  2. Substitute -1/2 for a: -10/2a becomes -10/2(-1/2)

Simplifying the Equation

Now that we have substituted the values, we can simplify the equation:

-10/2(-1/2) = -10/(-1) = 10

The equation simplifies to 10, which is the value of the y-coordinate of the vertex.

The Significance of the Y-Coordinate of the Vertex

The y-coordinate of the vertex gives the greatest possible weekly profit for the company. In other words, it represents the maximum value of the profit function, which is a critical factor in decision-making. By identifying the y-coordinate of the vertex, the company can determine the optimal level of production to maximize its profit.

Real-World Applications of the Vertex Form

The vertex form of a quadratic function has numerous real-world applications, including:

  • Finance: The vertex form is used to model the behavior of financial instruments, such as stocks and bonds, to determine the maximum or minimum value of the investment.
  • Engineering: The vertex form is used to design and optimize systems, such as bridges and buildings, to ensure maximum stability and minimum cost.
  • Economics: The vertex form is used to model the behavior of economic systems, such as supply and demand, to determine the optimal level of production and pricing.

Conclusion

In conclusion, the vertex form of a quadratic function is a powerful tool for analyzing and solving quadratic equations. By substituting values into the equation and simplifying it, we can identify the y-coordinate of the vertex, which gives the greatest possible weekly profit for the company. The vertex form has numerous real-world applications, including finance, engineering, and economics, and is an essential tool for decision-making in various industries.

Step-by-Step Guide to Substituting Values into the Equation

  1. Identify the values to be substituted: b = 10 and a = -1/2
  2. Substitute the values into the equation: -b/2a becomes -10/2(-1/2)
  3. Simplify the equation: -10/2(-1/2) = -10/(-1) = 10
  4. Identify the y-coordinate of the vertex: 10

Frequently Asked Questions

  1. What is the vertex form of a quadratic function? The vertex form of a quadratic function is a mathematical representation of a quadratic equation in the form of f(x) = a(x - h)^2 + k.
  2. What is the significance of the y-coordinate of the vertex? The y-coordinate of the vertex gives the greatest possible weekly profit for the company.
  3. What are the real-world applications of the vertex form? The vertex form has numerous real-world applications, including finance, engineering, and economics.

References

In our previous article, we explored the concept of the vertex form of a quadratic function and its significance in various real-world applications. In this article, we will address some of the most frequently asked questions about the vertex form of a quadratic function.

Q: What is the vertex form of a quadratic function?

A: The vertex form of a quadratic function is a mathematical representation of a quadratic equation in the form of f(x) = a(x - h)^2 + k, where:

  • f(x) is the function
  • a is the coefficient of the squared term
  • h is the x-coordinate of the vertex
  • k is the y-coordinate of the vertex

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

A: The y-coordinate of the vertex gives the greatest possible weekly profit for the company. In other words, it represents the maximum value of the profit function, which is a critical factor in decision-making.

Q: How do I find the y-coordinate of the vertex?

A: To find the y-coordinate of the vertex, you need to substitute the values of a, b, and c into the equation f(x) = a(x - h)^2 + k. Then, simplify the equation to find the value of k.

Q: What are the real-world applications of the vertex form?

A: The vertex form has numerous real-world applications, including:

  • Finance: The vertex form is used to model the behavior of financial instruments, such as stocks and bonds, to determine the maximum or minimum value of the investment.
  • Engineering: The vertex form is used to design and optimize systems, such as bridges and buildings, to ensure maximum stability and minimum cost.
  • Economics: The vertex form is used to model the behavior of economic systems, such as supply and demand, to determine the optimal level of production and pricing.

Q: How do I use the vertex form to solve quadratic equations?

A: To use the vertex form to solve quadratic equations, you need to follow these steps:

  1. Identify the values of a, b, and c in the quadratic equation.
  2. Substitute the values into the vertex form equation f(x) = a(x - h)^2 + k.
  3. Simplify the equation to find the value of k.
  4. Use the value of k to find the x-coordinate of the vertex.
  5. Use the x-coordinate of the vertex to find the y-coordinate of the vertex.

Q: What are some common mistakes to avoid when using the vertex form?

A: Some common mistakes to avoid when using the vertex form include:

  • Incorrectly identifying the values of a, b, and c: Make sure to identify the correct values of a, b, and c in the quadratic equation.
  • Incorrectly substituting the values into the vertex form equation: Make sure to substitute the values correctly into the vertex form equation.
  • Not simplifying the equation correctly: Make sure to simplify the equation correctly to find the value of k.

Q: How do I graph a quadratic function using the vertex form?

A: To graph a quadratic function using the vertex form, you need to follow these steps:

  1. Identify the values of a, b, and c in the quadratic equation.
  2. Substitute the values into the vertex form equation f(x) = a(x - h)^2 + k.
  3. Simplify the equation to find the value of k.
  4. Use the value of k to find the x-coordinate of the vertex.
  5. Use the x-coordinate of the vertex to find the y-coordinate of the vertex.
  6. Plot the vertex on the graph.
  7. Plot the x-intercepts on the graph.
  8. Plot the y-intercept on the graph.

Conclusion

In conclusion, the vertex form of a quadratic function is a powerful tool for analyzing and solving quadratic equations. By understanding the significance of the y-coordinate of the vertex and how to find it, you can use the vertex form to solve quadratic equations and graph quadratic functions. Remember to avoid common mistakes and follow the steps outlined above to ensure accurate results.

Frequently Asked Questions

  1. What is the vertex form of a quadratic function? The vertex form of a quadratic function is a mathematical representation of a quadratic equation in the form of f(x) = a(x - h)^2 + k.
  2. What is the significance of the y-coordinate of the vertex? The y-coordinate of the vertex gives the greatest possible weekly profit for the company.
  3. How do I find the y-coordinate of the vertex? To find the y-coordinate of the vertex, you need to substitute the values of a, b, and c into the equation f(x) = a(x - h)^2 + k and simplify the equation to find the value of k.
  4. What are the real-world applications of the vertex form? The vertex form has numerous real-world applications, including finance, engineering, and economics.

References