Which Equation Is $y=-3x^2-12x-2$ Rewritten In Vertex Form?A. $y=-3(x+2)^2+10$ B. $y=-3(x-2)^2+10$ C. $y=-3(x+2)^2-14$ D. $y=-3(x-2)^2-2$

Understanding Vertex Form

Vertex form is a way of expressing quadratic equations in the form of $y=a(x-h)^2+k$, 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 rewrite the quadratic equation $y=-3x^2-12x-2$ in vertex form.

The Process of Rewriting in Vertex Form

To rewrite a quadratic equation in vertex form, we need to complete the square. This involves manipulating the equation to create a perfect square trinomial. The general steps for rewriting a quadratic equation in vertex form are as follows:

1. Factor out the coefficient of the $x^2$ term.
2. Move the constant term to the right-hand side of the equation.
3. Add and subtract the square of half the coefficient of the $x$ term to the left-hand side of the equation.
4. Factor the left-hand side of the equation into a perfect square trinomial.
5. Write the equation in vertex form.

Applying the Process to the Given Equation

Let's apply the process to the given equation $y=-3x^2-12x-2$.

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

The coefficient of the $x^2$ term is -3. We can factor this out to get:

$$y=-3(x^2+4x)-2$$

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

We can move the constant term to the right-hand side of the equation by adding 2 to both sides:

$$y+2=-3(x^2+4x)$$

Step 3: Add and subtract the square of half the coefficient of the $x$ term to the left-hand side of the equation

The coefficient of the $x$ term is 4. Half of this is 2, and the square of 2 is 4. We can add and subtract 4 to the left-hand side of the equation:

$$y+2=-3(x^2+4x+4-4)$$

Step 4: Factor the left-hand side of the equation into a perfect square trinomial

We can factor the left-hand side of the equation into a perfect square trinomial:

$$y+2=-3((x+2)^2-4)$$

Step 5: Write the equation in vertex form

We can simplify the equation by distributing the -3:

$$y=-3(x+2)^2+12$$

However, we need to rewrite the equation in the exact form of the options given. To do this, we need to move the constant term to the right-hand side of the equation and subtract 12 from both sides:

$$y=-3(x+2)^2-2$$

Conclusion

In this article, we have explored how to rewrite the quadratic equation $y=-3x^2-12x-2$ in vertex form. We have applied the process of completing the square to rewrite the equation in the form of $y=a(x-h)^2+k$. The final answer is $y=-3(x+2)^2-2$.

Answer

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Q: What is vertex form?

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

Q: Why is vertex form useful?

A: Vertex form is particularly useful for identifying the vertex and the direction of the parabola. It can also be used to find the maximum or minimum value of a quadratic function.

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

A: To rewrite a quadratic equation in vertex form, you need to complete the square. This involves manipulating the equation to create a perfect square trinomial.

Q: What is completing the square?

A: Completing the square is a process of manipulating a quadratic equation to create a perfect square trinomial. This involves adding and subtracting a constant term to the left-hand side of the equation.

Q: How do I complete the square?

A: To complete the square, follow these steps:

1. Factor out the coefficient of the $x^2$ term.
2. Move the constant term to the right-hand side of the equation.
3. Add and subtract the square of half the coefficient of the $x$ term to the left-hand side of the equation.
4. Factor the left-hand side of the equation into a perfect square trinomial.
5. Write the equation in vertex form.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It is represented by the coordinates $(h,k)$ in the vertex form of a quadratic equation.

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

A: To find the vertex of a parabola, you need to rewrite the quadratic equation in vertex form. The vertex is represented by the coordinates $(h,k)$ in the vertex form.

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

A: The vertex of a parabola represents the maximum or minimum value of the quadratic function. It can also be used to determine the direction of the parabola.

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

A: Yes, you can use vertex form to graph a quadratic function. The vertex form provides a clear representation of the vertex and the direction of the parabola, making it easier to graph the function.

Q: Are there any other ways to rewrite a quadratic equation?

A: Yes, there are other ways to rewrite a quadratic equation, such as using the factoring method or the quadratic formula. However, vertex form is a particularly useful way to rewrite a quadratic equation when you need to identify the vertex and the direction of the parabola.

Conclusion

In this article, we have answered some of the most frequently asked questions about rewriting quadratic equations in vertex form. We have covered topics such as vertex form, completing the square, and the significance of the vertex of a parabola. Whether you are a student or a teacher, this article provides a comprehensive guide to rewriting quadratic equations in vertex form.