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

Introduction

In mathematics, the vertex form of a quadratic function is a powerful tool for representing and analyzing quadratic equations. It provides a convenient way to express quadratic functions in a form that highlights their key characteristics, such as the vertex and the direction of opening. In this article, we will explore the concept of vertex form and learn how to convert a quadratic function from standard form to vertex form.

What is Vertex Form?

Vertex form is a way of expressing a quadratic function in the form $f(x)=a(x-h)^2+k$, where $(h,k)$ is the vertex of the parabola. The vertex form is particularly useful for identifying the vertex, the axis of symmetry, and the direction of opening of a quadratic function.

Converting from 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 follow these steps:

1. Factor out the coefficient of $x^2$, which is $a$.
2. Take half of the coefficient of $x$ and square it.
3. Add and subtract the squared value to the expression.
4. Factor the expression into a perfect square trinomial.

Example: Converting $f(x)=x^2+6x+3$ to Vertex Form

Let's apply the steps above to convert the quadratic function $f(x)=x^2+6x+3$ to vertex form.

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

The coefficient of $x^2$ is 1, so we can write the expression as $f(x)=1(x^2+6x)+3$.

Step 2: Take half of the coefficient of $x$ and square it

The coefficient of $x$ is 6, so half of it is 3. Squaring 3 gives us 9.

Step 3: Add and subtract the squared value to the expression

We add and subtract 9 to the expression: $f(x)=1(x^2+6x+9-9)+3$.

Step 4: Factor the expression into a perfect square trinomial

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

Step 5: Simplify the expression

Finally, we simplify the expression by combining like terms: $f(x)=(x+3)^2-9+3$.

Step 6: Write the final expression in vertex form

The final expression in vertex form is $f(x)=(x+3)^2-6$.

Conclusion

In this article, we learned how to convert a quadratic function from standard form to vertex form using the method of completing the square. We applied this method to the quadratic function $f(x)=x^2+6x+3$ and obtained the equivalent expression in vertex form: $f(x)=(x+3)^2-6$. This demonstrates the power of vertex form in representing and analyzing quadratic functions.

Which Function in Vertex Form is Equivalent to $f(x)=x^2+6x+3$?

Based on our analysis, we can conclude that the correct answer is:

• B. $f(x)=(x+3)^2-6$

This is because we obtained the equivalent expression in vertex form as $f(x)=(x+3)^2-6$ by completing the square.

Discussion

• What is the vertex of the parabola represented by $f(x)=x^2+6x+3$?

  The vertex of the parabola is $(h,k)=(-3,-6)$.

• What is the axis of symmetry of the parabola represented by $f(x)=x^2+6x+3$?

  The axis of symmetry is the vertical line $x=-3$.

• What is the direction of opening of the parabola represented by $f(x)=x^2+6x+3$?

  The parabola opens upward.

Additional Examples

• Converting $f(x)=x^2-4x+4$ to Vertex Form

  We can convert the quadratic function $f(x)=x^2-4x+4$ to vertex form by completing the square:

  $f(x)=1(x^2-4x)+4$

  $f(x)=1(x^2-4x+4-4)+4$

  $f(x)=1((x-2)^2-4)+4$

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

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

  The final expression in vertex form is $f(x)=(x-2)^2$.

• Converting $f(x)=x^2+2x+1$ to Vertex Form

  We can convert the quadratic function $f(x)=x^2+2x+1$ to vertex form by completing the square:

  $f(x)=1(x^2+2x)+1$

  $f(x)=1(x^2+2x+1-1)+1$

  $f(x)=1((x+1)^2-1)+1$

  $f(x)=(x+1)^2-1+1$

  $f(x)=(x+1)^2$

  The final expression in vertex form is $f(x)=(x+1)^2$.

Conclusion

Q&A: Frequently Asked Questions about Vertex Form

Q: What is vertex form?

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

1. Factor out the coefficient of $x^2$, which is $a$.
2. Take half of the coefficient of $x$ and square it.
3. Add and subtract the squared value to the expression.
4. Factor the expression into a perfect square trinomial.

Q: What is the vertex of the parabola represented by $f(x)=x^2+6x+3$?

A: The vertex of the parabola is $(h,k)=(-3,-6)$.

Q: What is the axis of symmetry of the parabola represented by $f(x)=x^2+6x+3$?

A: The axis of symmetry is the vertical line $x=-3$.

Q: What is the direction of opening of the parabola represented by $f(x)=x^2+6x+3$?

A: The parabola opens upward.

Q: How do I find the vertex of a parabola represented by a quadratic function in standard form?

A: To find the vertex of a parabola represented by a quadratic function in standard form, you need to complete the square. The steps are:

1. Factor out the coefficient of $x^2$, which is $a$.
2. Take half of the coefficient of $x$ and square it.
3. Add and subtract the squared value to the expression.
4. Factor the expression into a perfect square trinomial.

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

A: The vertex of a parabola represents the minimum or maximum point of the parabola. It is the point where the parabola changes direction.

Q: How do I determine the direction of opening of a parabola?

A: To determine the direction of opening of a parabola, you need to look at the coefficient of the $x^2$ term. If the coefficient is positive, the parabola opens upward. If the coefficient is negative, the parabola opens downward.

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

A: Yes, you can convert a quadratic function from vertex form to standard form by expanding the squared term.

Q: How do I expand a squared term in vertex form?

A: To expand a squared term in vertex form, you need to multiply the term inside the parentheses by itself.

Q: What is the difference between vertex form and standard form?

A: The main difference between vertex form and standard form is the way the quadratic function is expressed. Vertex form highlights the vertex of the parabola, while standard form highlights the coefficients of the terms.

Q: When should I use vertex form and when should I use standard form?

A: You should use vertex form when you need to highlight the vertex of the parabola, such as when graphing a quadratic function. You should use standard form when you need to highlight the coefficients of the terms, such as when solving a quadratic equation.

Conclusion

In this article, we answered frequently asked questions about vertex form, including how to convert a quadratic function from standard form to vertex form, how to find the vertex of a parabola, and how to determine the direction of opening of a parabola. We also discussed the significance of the vertex of a parabola and the difference between vertex form and standard form.