Understanding the Given Quadratic Function

Oct 13, 2025 by miodeya.com 118 views

Introduction

Quadratic functions are a fundamental concept in mathematics, and they can be expressed in various forms. The vertex form of a quadratic function is a specific form that highlights the vertex of the parabola represented by the function. In this article, we will focus on rewriting the given quadratic function in vertex form.

Understanding the Given Quadratic Function

The given quadratic function is $f(x) = 2x^2 - 8x + 14$ . This function is in the standard form of a quadratic function, which is $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. In this case, $a = 2$ , $b = -8$ , and $c = 14$ .

The Vertex Form of a Quadratic Function

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 parabola represented by the function. To rewrite the given quadratic function in vertex form, we need to complete the square.

Completing the Square

To complete the square, we need to follow these steps:

Factor out the coefficient of the $x^2$ term, which is $2$ in this case.
Take half of the coefficient of the $x$ term, which is $-8$ in this case, and square it.
Add and subtract the squared value from the expression.

Step 1: Factor out the Coefficient of the $x^2$ Term

We start by factoring out the coefficient of the $x^2$ term, which is $2$ . This gives us:

$f(x) = 2(x^2 - 4x) + 14$

Step 2: Take Half of the Coefficient of the $x$ Term and Square It

Next, we take half of the coefficient of the $x$ term, which is $-8$ , and square it. This gives us:

$(-8/2)^2 = 16$

Step 3: Add and Subtract the Squared Value

We add and subtract the squared value from the expression:

$f(x) = 2(x^2 - 4x + 16 - 16) + 14$

Simplifying the Expression

We can simplify the expression by combining like terms:

$f(x) = 2(x^2 - 4x + 16) - 32 + 14$

$f(x) = 2(x^2 - 4x + 16) - 18$

Rewriting the Expression in Vertex Form

We can now rewrite the expression in vertex form by factoring the perfect square trinomial:

$f(x) = 2(x - 2)^2 - 18$

Conclusion

In this article, we have rewritten the given quadratic function in vertex form. We started by factoring out the coefficient of the $x^2$ term, taking half of the coefficient of the $x$ term and squaring it, and adding and subtracting the squared value. We then simplified the expression and factored the perfect square trinomial to obtain the vertex form of the quadratic function. The vertex form of the quadratic function is $f(x) = 2(x - 2)^2 - 18$ .

Final Answer

$The final answer is: $\boxed{2(x - 2)^2 - 18}$<br/> **Quadratic Function Vertex Form Q&A**$

Introduction

In our previous article, we rewrote the quadratic function $f(x) = 2x^2 - 8x + 14$ in vertex form. In this article, we will answer some frequently asked questions about quadratic functions and vertex form.

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 $(h, k)$ is the vertex of the parabola represented by the function.

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

A: To rewrite a quadratic function in vertex form, you need to complete the square. This involves factoring out the coefficient of the $x^2$ term, taking half of the coefficient of the $x$ term and squaring it, and adding and subtracting the squared value.

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

A: The vertex form of a quadratic function highlights the vertex of the parabola represented by the function. This is useful in various applications, such as graphing, optimization, and modeling real-world phenomena.

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 vertex form. The vertex is given by the point $(h, k)$ in the vertex form $f(x) = a(x - h)^2 + k$ .

Q: Can I use the vertex form of a quadratic function to graph the parabola?

A: Yes, you can use the vertex form of a quadratic function to graph the parabola. The vertex form gives you the coordinates of the vertex, which is the lowest or highest point on the parabola.

Q: How do I use the vertex form of a quadratic function to optimize a function?

A: To optimize a function using the vertex form, you need to find the vertex of the function. The vertex represents the maximum or minimum value of the function.

Q: Can I use the vertex form of a quadratic function to model real-world phenomena?

A: Yes, you can use the vertex form of a quadratic function to model real-world phenomena. The vertex form gives you a clear and concise representation of the function, which can be used to model various real-world situations.

Q: What are some common applications of the vertex form of a quadratic function?

A: Some common applications of the vertex form of a quadratic function include:

Graphing parabolas
Optimizing functions
Modeling real-world phenomena
Solving quadratic equations

Conclusion

In this article, we have answered some frequently asked questions about quadratic functions and vertex form. We have discussed the significance of the vertex form, how to rewrite a quadratic function in vertex form, and how to use the vertex form to graph the parabola, optimize a function, and model real-world phenomena.

Final Answer

The final answer is: $\boxed{2(x - 2)^2 - 18}$