Which Function In Vertex Form Is Equivalent To F ( X ) = X 2 + 6 X + 3 F(x) = X^2 + 6x + 3 F ( X ) = X 2 + 6 X + 3 ?A. F ( X ) = ( X + 3 ) 2 + 3 F(x) = (x + 3)^2 + 3 F ( X ) = ( X + 3 ) 2 + 3 B. F ( X ) = ( X + 3 ) 2 − 6 F(x) = (x + 3)^2 - 6 F ( X ) = ( X + 3 ) 2 − 6 C. F ( X ) = ( X + 6 ) 2 + 3 F(x) = (x + 6)^2 + 3 F ( X ) = ( X + 6 ) 2 + 3 D. F ( X ) = ( X + 6 ) 2 − 6 F(x) = (x + 6)^2 - 6 F ( X ) = ( X + 6 ) 2 − 6

Understanding Vertex Form

Vertex form is a way of expressing quadratic functions in the form of $f(x) = a(x - h)^2 + k$, where $(h, k)$ represents the vertex of the parabola. This form is particularly useful for identifying the vertex and the direction of the parabola. In this article, we will explore how to convert a quadratic function from standard form to vertex form.

Converting Standard Form to Vertex Form

To convert a quadratic function from standard form to vertex form, we need to complete the square. The standard form of a quadratic function is $f(x) = ax^2 + bx + c$. To convert it to vertex form, we need to rewrite it in the form of $f(x) = a(x - h)^2 + k$.

Let's consider the given quadratic function $f(x) = x^2 + 6x + 3$. Our goal is to rewrite it in vertex form.

Step 1: Factor out the coefficient of $x^2$

The first step is to factor out the coefficient of $x^2$, which is 1 in this case. We can write the function as $f(x) = 1(x^2 + 6x) + 3$.

Step 2: Add and subtract the square of half the coefficient of $x$

Next, we need to add and subtract the square of half the coefficient of $x$. The coefficient of $x$ is 6, so half of it is 3. The square of 3 is 9. We can add and subtract 9 inside the parentheses: $f(x) = 1(x^2 + 6x + 9 - 9) + 3$.

Step 3: Factor the perfect square trinomial

Now, we can factor the perfect square trinomial: $f(x) = 1((x + 3)^2 - 9) + 3$.

Step 4: Simplify the expression

Finally, we can simplify the expression by distributing the 1: $f(x) = (x + 3)^2 - 9 + 3$.

Simplifying Further

We can simplify the expression further by combining the constants: $f(x) = (x + 3)^2 - 6$.

Conclusion

In conclusion, the function in vertex form that is equivalent to $f(x) = x^2 + 6x + 3$ is $f(x) = (x + 3)^2 - 6$.

Answer

The correct answer is B. $f(x) = (x + 3)^2 - 6$.

Why is this the correct answer?

This is the correct answer because we have successfully converted the quadratic function from standard form to vertex form using the steps outlined above. The vertex form of the function is $f(x) = (x + 3)^2 - 6$, which matches option B.

What is the significance of vertex form?

Vertex form is a powerful tool for identifying the vertex and the direction of a parabola. It is particularly useful for graphing quadratic functions and for solving problems involving quadratic equations. By converting a quadratic function from standard form to vertex form, we can gain a deeper understanding of the function's behavior and make it easier to work with.

How to use vertex form in real-world applications

Vertex form has many real-world applications, including:

• Graphing quadratic functions: Vertex form makes it easy to graph quadratic functions by identifying the vertex and the direction of the parabola.
• Solving quadratic equations: Vertex form can be used to solve quadratic equations by finding the x-coordinate of the vertex.
• Optimization problems: Vertex form can be used to solve optimization problems by finding the maximum or minimum value of a quadratic function.

Conclusion

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Q&A: Vertex Form and Quadratic Functions

Q: What is vertex form?

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

Q: Why is vertex form important?

A: Vertex form is important because it makes it easy to identify the vertex and the direction of a parabola. It is particularly useful for graphing quadratic functions and for solving problems involving quadratic equations.

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. The steps are:

1. Factor out the coefficient of $x^2$.
2. Add and subtract the square of half the coefficient of $x$.
3. Factor the perfect square trinomial.
4. Simplify the expression.

Q: What is the significance of the vertex in vertex form?

A: The vertex in vertex form represents the minimum or maximum value of the quadratic function. It is the point on the parabola where the function changes direction.

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

A: To find the vertex of a quadratic function in vertex form, you need to identify the values of $h$ and $k$. The vertex is represented by the point $(h, k)$.

Q: Can I use vertex form to solve quadratic equations?

A: Yes, you can use vertex form to solve quadratic equations. By finding the x-coordinate of the vertex, you can solve for the value of $x$.

Q: What are some real-world applications of vertex form?

A: Some real-world applications of vertex form include:

• Graphing quadratic functions
• Solving quadratic equations
• Optimization problems
• Physics and engineering problems

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

A: To graph a quadratic function in vertex form, you need to identify the vertex and the direction of the parabola. You can use the vertex to draw the graph of the function.

Q: Can I use vertex form to find the maximum or minimum value of a quadratic function?

A: Yes, you can use vertex form to find the maximum or minimum value of a quadratic function. The vertex represents the maximum or minimum value of the function.

Q: How do I convert a quadratic function from vertex form to standard form?

A: To convert a quadratic function from vertex form to standard form, you need to expand the squared term and simplify the expression.

Q: What are some common mistakes to avoid when working with vertex form?

A: Some common mistakes to avoid when working with vertex form include:

• Not completing the square correctly
• Not identifying the vertex correctly
• Not using the correct values for $h$ and $k$

Conclusion

In conclusion, vertex form is a powerful tool for working with quadratic functions. By understanding the concepts and techniques outlined in this article, you can gain a deeper understanding of quadratic functions and make it easier to work with them.