The First Steps In Writing F ( X ) = 4 X 2 + 48 X + 10 Π F(x) = 4x^2 + 48x + 10\pi F ( X ) = 4 X 2 + 48 X + 10 Π In Vertex Form Are Shown: F ( X ) = 4 ( X 2 + 12 X ) + 10 F(x) = 4(x^2 + 12x) + 10 F ( X ) = 4 ( X 2 + 12 X ) + 10 ( 12 2 ) 2 = 36 \left(\frac{12}{2}\right)^2 = 36 ( 2 12 ​ ) 2 = 36 What Is The Function Written In Vertex Form?A. F ( X ) = 4 ( X + 6 ) 2 + 10 F(x) = 4(x + 6)^2 + 10 F ( X ) = 4 ( X + 6 ) 2 + 10 B.

Introduction

Writing a quadratic function in vertex form is an essential skill in mathematics, particularly in algebra and calculus. The vertex form of a quadratic function is a way of expressing the function in the form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this article, we will explore the first steps in writing the quadratic function $f(x) = 4x^2 + 48x + 10\pi$ in vertex form.

Understanding the Vertex Form

The vertex form of a quadratic function is a way of expressing the function in the form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The vertex form is useful because it allows us to easily identify the vertex of the parabola, which is the maximum or minimum point of the function.

Completing the Square

To write the quadratic function $f(x) = 4x^2 + 48x + 10\pi$ in vertex form, we need to complete the square. Completing the square involves rewriting the quadratic function in the form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Step 1: Factor Out the Coefficient of $x^2$

The first step in completing the square is to factor out the coefficient of $x^2$. In this case, the coefficient of $x^2$ is 4, so we can factor it out as follows:

$f(x) = 4(x^2 + 12x) + 10\pi$

Step 2: Add and Subtract the Square of Half the Coefficient of $x$

The next step is to add and subtract the square of half the coefficient of $x$. In this case, the coefficient of $x$ is 12, so half of it is 6. The square of 6 is 36, so we can add and subtract 36 as follows:

$f(x) = 4(x^2 + 12x + 36 - 36) + 10\pi$

Step 3: Rewrite the Function in Vertex Form

Now that we have added and subtracted 36, we can rewrite the function in vertex form as follows:

$f(x) = 4(x^2 + 12x + 36) - 144 + 10\pi$

$f(x) = 4(x + 6)^2 - 144 + 10\pi$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

Simplifying the Function

The final step is to simplify the function by combining like terms. In this case, we can combine the constants as follows:

$f(x) = 4(x + 6)^2 - 144 + 10\pi$

$f(x) = 4(x + 6)^2 - 144 + 10\pi$

$f(x) = 4(x + 6)^2 - 144 + 10\pi$

$f(x) = 4(x + 6)^2 - 144 + 10\pi$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

$f(x) = 4(x + 6)^2 + 10\pi - 144$

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

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

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

A: To write a quadratic function in vertex form, you need to complete the square. This involves rewriting the quadratic function in the form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Q: What is completing the square?

A: Completing the square is a process of rewriting a quadratic function in the form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. This involves adding and subtracting a constant term to create a perfect square trinomial.

Q: How do I complete the square?

A: To complete the square, you need to follow these steps:

1. Factor out the coefficient of $x^2$.
2. Add and subtract the square of half the coefficient of $x$.
3. Rewrite the function in vertex form.

Q: What is the formula for completing the square?

A: The formula for completing the square is:

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

where $(h, k)$ is the vertex of the parabola.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you need to use the formula:

$h = -\frac{b}{2a}$

where $a$ and $b$ are the coefficients of the quadratic function.

Q: What is the significance of the vertex of a parabola?

A: The vertex of a parabola is the maximum or minimum point of the function. It is the point where the function changes from increasing to decreasing or vice versa.

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

A: The vertex form of a quadratic function is useful for graphing the function and finding the maximum or minimum point. It is also useful for solving quadratic equations.

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

A: The vertex form of a quadratic function has many applications in mathematics and science. Some common applications include:

• Graphing quadratic functions
• Finding the maximum or minimum point of a quadratic function
• Solving quadratic equations
• Modeling real-world phenomena, such as the trajectory of a projectile or the motion of an object under the influence of gravity.

Conclusion

In conclusion, the vertex form of a quadratic function is a powerful tool for graphing and solving quadratic equations. By completing the square, we can rewrite a quadratic function in the form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. This allows us to easily identify the maximum or minimum point of the function and solve quadratic equations.