What Is The Vertex Of The Graph Of The Function $f(x)=x^2+8x-2$?A. $(-8,-2$\] B. $(0,-2$\] C. $(-4,18$\] D. $(-4,-18$\]

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Understanding the Concept of Vertex

The vertex of a quadratic function is the highest or lowest point on the graph of the function. It is a crucial concept in mathematics, particularly in algebra and calculus. The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ represents the coordinates of the vertex. In this article, we will explore how to find the vertex of the graph of the function $f(x) = x^2 + 8x - 2$.

Finding the Vertex of a Quadratic Function

To find the vertex of a quadratic function, we can use the method of completing the square. This method involves rewriting the quadratic function in the form $f(x) = a(x - h)^2 + k$. The vertex of the function is then given by the coordinates $(h, k)$.

Step 1: Rewrite the Quadratic Function

The given quadratic function is $f(x) = x^2 + 8x - 2$. To rewrite this function in the form $f(x) = a(x - h)^2 + k$, we need to complete the square.

Completing the Square

To complete the square, we start by taking half of the coefficient of the $x$ term, which is $8$, and squaring it. This gives us $(8/2)^2 = 16$. We then add and subtract $16$ inside the parentheses to maintain the value of the expression.

Rewriting the Function

The rewritten function is $f(x) = (x^2 + 8x + 16) - 16 - 2$. We can simplify this expression by combining like terms.

Simplifying the Expression

The simplified expression is $f(x) = (x + 4)^2 - 18$. This is the vertex form of the quadratic function.

Identifying the Vertex

The vertex of the function is given by the coordinates $(h, k)$. In this case, $h = -4$ and $k = -18$. Therefore, the vertex of the graph of the function $f(x) = x^2 + 8x - 2$ is $(-4, -18)$.

Conclusion

In this article, we have explored how to find the vertex of the graph of the function $f(x) = x^2 + 8x - 2$. We used the method of completing the square to rewrite the function in the form $f(x) = a(x - h)^2 + k$. The vertex of the function is then given by the coordinates $(h, k)$. We have shown that the vertex of the graph of the function $f(x) = x^2 + 8x - 2$ is $(-4, -18)$.

Answer

The correct answer is D. $(-4, -18)$.

Discussion

The vertex of a quadratic function is an important concept in mathematics. It is used to determine the maximum or minimum value of the function. In this article, we have shown how to find the vertex of the graph of the function $f(x) = x^2 + 8x - 2$ using the method of completing the square. We have also identified the vertex of the function as $(-4, -18)$.

Related Topics

• Quadratic Functions: Quadratic functions are polynomial functions of degree two. They have the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
• Completing the Square: Completing the square is a method used to rewrite a quadratic function in the form $f(x) = a(x - h)^2 + k$. It involves adding and subtracting a constant term inside the parentheses to maintain the value of the expression.
• Vertex Form: The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$. The vertex of the function is then given by the coordinates $(h, k)$.

Applications

The concept of vertex is used in various applications, including:

• Optimization: The vertex of a quadratic function can be used to determine the maximum or minimum value of the function.
• Graphing: The vertex of a quadratic function can be used to graph the function.
• Engineering: The vertex of a quadratic function can be used to design and optimize systems.

Conclusion

In conclusion, the vertex of a quadratic function is an important concept in mathematics. It is used to determine the maximum or minimum value of the function. In this article, we have shown how to find the vertex of the graph of the function $f(x) = x^2 + 8x - 2$ using the method of completing the square. We have also identified the vertex of the function as $(-4, -18)$.

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Understanding the Concept of Vertex

The vertex of a quadratic function is the highest or lowest point on the graph of the function. It is a crucial concept in mathematics, particularly in algebra and calculus. The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ represents the coordinates of the vertex. In this article, we will explore how to find the vertex of the graph of the function $f(x) = x^2 + 8x - 2$.

Q&A: Finding the Vertex of a Quadratic Function

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

A: The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ represents the coordinates of the vertex.

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

A: To find the vertex of a quadratic function, you can use the method of completing the square. This method involves rewriting the quadratic function in the form $f(x) = a(x - h)^2 + k$. The vertex of the function is then given by the coordinates $(h, k)$.

Q: What is completing the square?

A: Completing the square is a method used to rewrite a quadratic function in the form $f(x) = a(x - h)^2 + k$. It involves adding and subtracting a constant term inside the parentheses to maintain the value of the expression.

Q: How do I complete the square?

A: To complete the square, you start by taking half of the coefficient of the $x$ term, which is $8$, and squaring it. This gives us $(8/2)^2 = 16$. We then add and subtract $16$ inside the parentheses to maintain the value of the expression.

Q: What is the vertex of the graph of the function $f(x) = x^2 + 8x - 2$?

A: The vertex of the graph of the function $f(x) = x^2 + 8x - 2$ is $(-4, -18)$.

Q: Why is the vertex of a quadratic function important?

A: The vertex of a quadratic function is important because it represents the maximum or minimum value of the function. It is used in various applications, including optimization, graphing, and engineering.

Q: How do I use the vertex of a quadratic function in real-world applications?

A: The vertex of a quadratic function can be used in various real-world applications, including optimization, graphing, and engineering. For example, it can be used to design and optimize systems, such as bridges and buildings.

Conclusion

In conclusion, the vertex of a quadratic function is an important concept in mathematics. It is used to determine the maximum or minimum value of the function. In this article, we have shown how to find the vertex of the graph of the function $f(x) = x^2 + 8x - 2$ using the method of completing the square. We have also identified the vertex of the function as $(-4, -18)$. We hope that this article has provided you with a better understanding of the concept of vertex and how to use it in real-world applications.

Related Topics

• Quadratic Functions: Quadratic functions are polynomial functions of degree two. They have the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
• Completing the Square: Completing the square is a method used to rewrite a quadratic function in the form $f(x) = a(x - h)^2 + k$. It involves adding and subtracting a constant term inside the parentheses to maintain the value of the expression.
• Vertex Form: The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$. The vertex of the function is then given by the coordinates $(h, k)$.

Applications

The concept of vertex is used in various applications, including:

• Optimization: The vertex of a quadratic function can be used to determine the maximum or minimum value of the function.
• Graphing: The vertex of a quadratic function can be used to graph the function.
• Engineering: The vertex of a quadratic function can be used to design and optimize systems.

Conclusion

In conclusion, the vertex of a quadratic function is an important concept in mathematics. It is used to determine the maximum or minimum value of the function. In this article, we have shown how to find the vertex of the graph of the function $f(x) = x^2 + 8x - 2$ using the method of completing the square. We have also identified the vertex of the function as $(-4, -18)$. We hope that this article has provided you with a better understanding of the concept of vertex and how to use it in real-world applications.