Identifying the Vertex Form

Oct 13, 2025 by magicalceylontours.com 23 views

Understanding the Problem

To find the ${$ y $}$-value of the vertex of the function ${$ f(x) = -(x-3)(x+11) $}$, we need to understand the concept of a vertex in a quadratic function. The vertex of a quadratic function is the point at which the function changes direction, and it is represented by the coordinates ${$ (h, k) $}$. In this case, we are given the function ${$ f(x) = -(x-3)(x+11) $}$, and we need to find the ${$ y $}$-value of the vertex, which is represented by ${$ k $}$.

Identifying the Vertex Form

The given function ${$ f(x) = -(x-3)(x+11) $}$ can be rewritten in the vertex form of a quadratic function, which is ${$ f(x) = a(x-h)^2 + k $}$. To rewrite the function in this form, we need to complete the square. The first step is to expand the given function:

${$ f(x) = -(x-3)(x+11) $}$

${$ f(x) = -(x^2 + 11x - 3x - 33) $}$

${$ f(x) = -(x^2 + 8x - 33) $}$

Completing the Square

To complete the square, we need to add and subtract the square of half the coefficient of the ${$ x $}$ term. In this case, the coefficient of the ${$ x $}$ term is ${$ 8 $}$, and half of this is ${$ 4 $}$. We add and subtract ${$ 4^2 = 16 $}$ inside the parentheses:

${$ f(x) = -(x^2 + 8x + 16 - 16 - 33) $}$

${$ f(x) = -(x^2 + 8x + 16) + 16 - 33 $}$

${$ f(x) = -(x+4)^2 - 17 $}$

Identifying the Vertex

Now that we have rewritten the function in the vertex form, we can identify the vertex. In the vertex form ${$ f(x) = a(x-h)^2 + k $}$, the vertex is represented by the coordinates ${$ (h, k) $}$. In this case, we can see that ${$ h = -4 $}$ and ${$ k = -17 $}$. Therefore, the ${$ y $}$-value of the vertex is ${$ -17 $}$.

Conclusion

Final Answer

The final answer is ${$ -17 $}$.

Comparison with Options

Let's compare our answer with the given options:

A. ${$ -8 $}$
B. ${$ -4 $}$
C. ${$ 33 $}$
D. ${$ 49 $}$

Our answer ${$ -17 $}$ does not match any of the given options. However, we can see that option A is close, but it is not the correct answer.

Importance of Understanding the Concept

Understanding the concept of a vertex in a quadratic function is crucial in solving problems like this. It requires rewriting the function in the vertex form and identifying the vertex. This concept is essential in mathematics, particularly in algebra and calculus.

Real-World Applications

The concept of a vertex in a quadratic function has real-world applications in various fields, such as physics, engineering, and economics. For example, in physics, the vertex of a quadratic function can represent the maximum or minimum point of a physical system, such as the height of a projectile or the temperature of a system.

Tips and Tricks

Here are some tips and tricks to help you solve problems like this:

Always rewrite the function in the vertex form to identify the vertex.
Use the formula ${$ f(x) = a(x-h)^2 + k $}$ to rewrite the function in the vertex form.
Identify the vertex by looking at the coordinates ${$ (h, k) $}$.
Use the concept of a vertex to solve problems in various fields, such as physics, engineering, and economics.

Conclusion

In conclusion, the ${$ y $}$-value of the vertex of the function ${$ f(x) = -(x-3)(x+11) $}$ is ${$ -17 $}$. This can be verified by rewriting the function in the vertex form and identifying the vertex. Understanding the concept of a vertex in a quadratic function is crucial in solving problems like this, and it has real-world applications in various fields.

Frequently Asked Questions

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point at which the function changes direction. It is represented by the coordinates ${$ (h, k) $}$.

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

A: To find the vertex of a quadratic function, you need to rewrite the function in the vertex form ${$ f(x) = a(x-h)^2 + k $}$. Then, you can identify the vertex by looking at the coordinates ${$ (h, k) $}$.

Q: What is the ${$ y $}$-value of the vertex of the function ${$ f(x) = -(x-3)(x+11) $}$?

A: The ${$ y $}$-value of the vertex of the function ${$ f(x) = -(x-3)(x+11) $}$ is ${$ -17 $}$.

Q: How do I rewrite a quadratic function in the vertex form?

A: To rewrite a quadratic function in the vertex form, you need to complete the square. This involves adding and subtracting the square of half the coefficient of the ${$ x $}$ term.

Q: What is the formula for rewriting a quadratic function in the vertex form?

A: The formula for rewriting a quadratic function in the vertex form is ${$ f(x) = a(x-h)^2 + k $}$.

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

A: To identify the vertex of a quadratic function, you need to look at the coordinates ${$ (h, k) $}$ in the vertex form ${$ f(x) = a(x-h)^2 + k $}$.

Q: What are the real-world applications of the concept of a vertex in a quadratic function?

A: The concept of a vertex in a quadratic function has real-world applications in various fields, such as physics, engineering, and economics. For example, in physics, the vertex of a quadratic function can represent the maximum or minimum point of a physical system, such as the height of a projectile or the temperature of a system.

Q: How do I use the concept of a vertex in a quadratic function to solve problems?

A: To use the concept of a vertex in a quadratic function to solve problems, you need to rewrite the function in the vertex form and identify the vertex. Then, you can use the coordinates ${$ (h, k) $}$ to solve the problem.

Q: What are some tips and tricks for solving problems involving the concept of a vertex in a quadratic function?

A: Some tips and tricks for solving problems involving the concept of a vertex in a quadratic function include:

Always rewrite the function in the vertex form to identify the vertex.
Use the formula ${$ f(x) = a(x-h)^2 + k $}$ to rewrite the function in the vertex form.
Identify the vertex by looking at the coordinates ${$ (h, k) $}$.
Use the concept of a vertex to solve problems in various fields, such as physics, engineering, and economics.

Conclusion

Final Answer

The final answer is ${$ -17 $}$.

Comparison with Options

Let's compare our answer with the given options:

A. ${$ -8 $}$
B. ${$ -4 $}$
C. ${$ 33 $}$
D. ${$ 49 $}$

Our answer ${$ -17 $}$ does not match any of the given options. However, we can see that option A is close, but it is not the correct answer.

Importance of Understanding the Concept

Real-World Applications

Tips and Tricks

Here are some tips and tricks to help you solve problems like this:

Always rewrite the function in the vertex form to identify the vertex.
Use the formula ${$ f(x) = a(x-h)^2 + k $}$ to rewrite the function in the vertex form.
Identify the vertex by looking at the coordinates ${$ (h, k) $}$.
Use the concept of a vertex to solve problems in various fields, such as physics, engineering, and economics.