Select The Correct Answer.Find The Vertex Of The Quadratic Function Given Below. F ( X ) = ( X − 4 ) ( X + 2 F(x) = (x-4)(x+2 F ( X ) = ( X − 4 ) ( X + 2 ]A. ( − 1 , 9 (-1, 9 ( − 1 , 9 ] B. ( − 4 , 2 (-4, 2 ( − 4 , 2 ] C. ( 1 , − 9 (1, -9 ( 1 , − 9 ] D. ( 4 , − 2 (4, -2 ( 4 , − 2 ]

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Introduction

In mathematics, 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. One of the important properties of a quadratic function is its vertex, which is the maximum or minimum point of the function. In this article, we will learn how to find the vertex of a quadratic function given in the form f(x) = (x - p)(x - q).

Understanding the Vertex Form

The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the function. To find the vertex, we need to rewrite the given function in the vertex form. Let's start with the given function f(x) = (x - 4)(x + 2).

Rewriting the Function in Vertex Form

To rewrite the function in vertex form, we need to expand the given function.

f(x) = (x - 4)(x + 2) f(x) = x^2 + 2x - 4x - 8 f(x) = x^2 - 2x - 8

Now, we need to complete the square to rewrite the function in vertex form.

f(x) = x^2 - 2x - 8 f(x) = (x^2 - 2x + 1) - 1 - 8 f(x) = (x - 1)^2 - 9

Finding the Vertex

Now that we have rewritten the function in vertex form, we can easily find the vertex. The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the function. Comparing this with our rewritten function, we can see that h = 1 and k = -9.

Conclusion

In this article, we learned how to find the vertex of a quadratic function given in the form f(x) = (x - p)(x - q). We started by rewriting the function in vertex form and then found the vertex by comparing the rewritten function with the vertex form. The vertex of the given function is (1, -9).

Answer

The correct answer is C. (1,−9)(1, -9).

Discussion

  • What is the vertex form of a quadratic function?
  • How do you rewrite a quadratic function in vertex form?
  • What is the vertex of the given function f(x) = (x - 4)(x + 2)?

Solutions

  • The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the function.
  • To rewrite a quadratic function in vertex form, you need to expand the function and then complete the square.
  • The vertex of the given function f(x) = (x - 4)(x + 2) is (1, -9).
    Quadratic Function Vertex: Frequently Asked Questions =====================================================

Introduction

In our previous article, we learned how to find the vertex of a quadratic function given in the form f(x) = (x - p)(x - q). In this article, we will answer some frequently asked questions related to quadratic function vertices.

Q&A

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

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

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

A: To rewrite a quadratic function in vertex form, you need to expand the function and then complete the square.

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

A: The vertex of a quadratic function is the maximum or minimum point of the function. It is an important concept in mathematics and has many real-world applications.

Q: How do you find the vertex of a quadratic function given in the form f(x) = ax^2 + bx + c?

A: To find the vertex of a quadratic function given in the form f(x) = ax^2 + bx + c, you need to rewrite the function in vertex form by completing the square.

Q: What is the relationship between the coefficients of a quadratic function and its vertex?

A: The coefficients of a quadratic function determine the position and shape of its vertex. The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where a, h, and k are related to the coefficients of the function.

Q: Can you give an example of finding the vertex of a quadratic function?

A: Let's consider the quadratic function f(x) = (x - 4)(x + 2). To find the vertex, we need to rewrite the function in vertex form by expanding and completing the square.

f(x) = (x - 4)(x + 2) f(x) = x^2 + 2x - 4x - 8 f(x) = x^2 - 2x - 8 f(x) = (x^2 - 2x + 1) - 1 - 8 f(x) = (x - 1)^2 - 9

Now that we have rewritten the function in vertex form, we can easily find the vertex. The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the function. Comparing this with our rewritten function, we can see that h = 1 and k = -9.

Q: What are some real-world applications of quadratic function vertices?

A: Quadratic function vertices have many real-world applications, including:

  • Modeling population growth and decline
  • Describing the motion of objects under the influence of gravity
  • Analyzing the behavior of electrical circuits
  • Optimizing business processes and resource allocation

Conclusion

In this article, we answered some frequently asked questions related to quadratic function vertices. We covered topics such as the vertex form of a quadratic function, rewriting a quadratic function in vertex form, and the significance of the vertex of a quadratic function. We also provided an example of finding the vertex of a quadratic function and discussed some real-world applications of quadratic function vertices.

Further Reading

Practice Problems

  • Find the vertex of the quadratic function f(x) = (x - 3)(x + 1).
  • Rewrite the quadratic function f(x) = x^2 + 4x + 4 in vertex form.
  • Find the vertex of the quadratic function f(x) = 2x^2 - 6x - 8.

Answers

  • The vertex of the quadratic function f(x) = (x - 3)(x + 1) is (1, -4).
  • The vertex form of the quadratic function f(x) = x^2 + 4x + 4 is f(x) = (x + 2)^2 - 4.
  • The vertex of the quadratic function f(x) = 2x^2 - 6x - 8 is (-1, -12).