Function $f(x) = X^2 - 2x - 6$ Is Written In Standard Form.What Is This Function Written In Vertex Form?A. $f(x) = (x - 1)^2 - 7$B. $f(x) = (x + 1)^2 - 7$C. $f(x) = (x - 1)^2 - 5$D. $f(x) = (x + 1)^2 - 5$

Introduction

In mathematics, quadratic functions are a fundamental concept in algebra and calculus. A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The standard form of a quadratic function is given by $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. However, it is often more convenient to express a quadratic function in vertex form, which is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this article, we will discuss how to convert a quadratic function from standard form to vertex form.

The Standard Form of a Quadratic Function

The standard form of a quadratic function is given by $f(x) = ax^2 + bx + c$. This form is useful for finding the roots of the function, but it does not provide any information about the vertex of the parabola. To find the vertex, we need to convert the function to vertex form.

Converting a Quadratic Function from Standard Form to Vertex Form

To convert a quadratic function from standard form to vertex form, we need to complete the square. This involves rewriting the function in a way that allows us to easily identify the vertex. The general steps for completing the square are as follows:

1. Write the function in standard form: Write the quadratic function in the form $f(x) = ax^2 + bx + c$.
2. Move the constant term to the right-hand side: Move the constant term $c$ to the right-hand side of the equation by subtracting it from both sides.
3. Take half of the coefficient of the linear term: Take half of the coefficient of the linear term $b$ and square it.
4. Add and subtract the squared term: Add and subtract the squared term to the right-hand side of the equation.
5. Factor the perfect square trinomial: Factor the perfect square trinomial on the left-hand side of the equation.

Example: Converting the Function $f(x) = x^2 - 2x - 6$ to Vertex Form

Let's use the function $f(x) = x^2 - 2x - 6$ as an example. We will follow the steps above to convert this function to vertex form.

Step 1: Write the function in standard form

The function is already in standard form: $f(x) = x^2 - 2x - 6$.

Step 2: Move the constant term to the right-hand side

Subtract $-6$ from both sides: $f(x) = x^2 - 2x = 6$.

Step 3: Take half of the coefficient of the linear term

Take half of the coefficient of the linear term $-2$: $-2/2 = -1$.

Step 4: Add and subtract the squared term

Add and subtract the squared term $(-1)^2 = 1$ to the right-hand side: $f(x) = x^2 - 2x + 1 - 1 = 6$.

Step 5: Factor the perfect square trinomial

Factor the perfect square trinomial on the left-hand side: $f(x) = (x - 1)^2 - 1 = 6$.

Step 6: Simplify the equation

Add $1$ to both sides: $f(x) = (x - 1)^2 = 7$.

Step 7: Write the function in vertex form

Write the function in vertex form: $f(x) = (x - 1)^2 - 7$.

Conclusion

In this article, we discussed how to convert a quadratic function from standard form to vertex form. We used the function $f(x) = x^2 - 2x - 6$ as an example and followed the steps for completing the square to convert it to vertex form. The vertex form of the function is $f(x) = (x - 1)^2 - 7$. This form is useful for identifying the vertex of the parabola and for graphing the function.

Answer

$$

Introduction

In our previous article, we discussed how to convert a quadratic function from standard form to vertex form. In this article, we will provide a Q&A guide to help you understand the process and apply it to different types of quadratic functions.

Q&A Guide

Q: What is the standard form of a quadratic function? A: The standard form of a quadratic function is given by $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

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.

Q: How do I convert a quadratic function from standard form to vertex form? A: To convert a quadratic function from standard form to vertex form, you need to complete the square. This involves rewriting the function in a way that allows you to easily identify the vertex.

Q: What are the steps for completing the square? A: The steps for completing the square are:

1. Write the function in standard form.
2. Move the constant term to the right-hand side.
3. Take half of the coefficient of the linear term and square it.
4. Add and subtract the squared term to the right-hand side.
5. Factor the perfect square trinomial.

Q: Can you provide an example of converting a quadratic function from standard form to vertex form? A: Let's use the function $f(x) = x^2 - 2x - 6$ as an example. We will follow the steps above to convert this function to vertex form.

Step 1: Write the function in standard form

The function is already in standard form: $f(x) = x^2 - 2x - 6$.

Step 2: Move the constant term to the right-hand side

Subtract $-6$ from both sides: $f(x) = x^2 - 2x = 6$.

Step 3: Take half of the coefficient of the linear term

Take half of the coefficient of the linear term $-2$: $-2/2 = -1$.

Step 4: Add and subtract the squared term

Add and subtract the squared term $(-1)^2 = 1$ to the right-hand side: $f(x) = x^2 - 2x + 1 - 1 = 6$.

Step 5: Factor the perfect square trinomial

Factor the perfect square trinomial on the left-hand side: $f(x) = (x - 1)^2 - 1 = 6$.

Step 6: Simplify the equation

Add $1$ to both sides: $f(x) = (x - 1)^2 = 7$.

Step 7: Write the function in vertex form

Write the function in vertex form: $f(x) = (x - 1)^2 - 7$.

Q: What is the vertex of the parabola in the example above? A: The vertex of the parabola is $(1, -7)$.

Q: How do I find the vertex of a parabola in vertex form? A: To find the vertex of a parabola in vertex form, you can simply read the coordinates of the vertex from the function.

Q: Can you provide another example of converting a quadratic function from standard form to vertex form? A: Let's use the function $f(x) = x^2 + 4x + 4$ as another example. We will follow the steps above to convert this function to vertex form.

Step 1: Write the function in standard form

The function is already in standard form: $f(x) = x^2 + 4x + 4$.

Step 2: Move the constant term to the right-hand side

Subtract $4$ from both sides: $f(x) = x^2 + 4x = -4$.

Step 3: Take half of the coefficient of the linear term

Take half of the coefficient of the linear term $4$: $4/2 = 2$.

Step 4: Add and subtract the squared term

Add and subtract the squared term $2^2 = 4$ to the right-hand side: $f(x) = x^2 + 4x + 4 - 4 = -4$.

Step 5: Factor the perfect square trinomial

Factor the perfect square trinomial on the left-hand side: $f(x) = (x + 2)^2 - 4 = -4$.

Step 6: Simplify the equation

Add $4$ to both sides: $f(x) = (x + 2)^2 = 0$.

Step 7: Write the function in vertex form

Write the function in vertex form: $f(x) = (x + 2)^2$.

Q: What is the vertex of the parabola in the example above? A: The vertex of the parabola is $(-2, 0)$.

Conclusion

In this article, we provided a Q&A guide to help you understand the process of converting a quadratic function from standard form to vertex form. We also provided two examples of converting quadratic functions from standard form to vertex form. By following the steps and examples provided, you should be able to convert any quadratic function from standard form to vertex form.