Which Equation Is \[$ Y = 6x^2 + 12x - 10 \$\] Rewritten In Vertex Form?A. \[$ Y = 6(x+1)^2 - 11 \$\] B. \[$ Y = 6(x+1)^2 - 10 \$\] C. \[$ Y = 6(x+1)^2 - 4 \$\] D. \[$ Y = 6(x+1)^2 - 16 \$\]

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 coordinates of the vertex of the parabola. This form is particularly useful for identifying the vertex and the direction of the parabola's opening.

The Given Equation

The given equation is $y = 6x^2 + 12x - 10$. To rewrite this equation in vertex form, we need to complete the square.

Completing the Square

To complete the square, we start by factoring out the coefficient of $x^2$, which is 6.

$$y = 6(x^2 + 2x) - 10$$

Next, we add and subtract the square of half the coefficient of $x$ inside the parentheses.

$$y = 6(x^2 + 2x + 1 - 1) - 10$$

Now, we can rewrite the equation as:

$$y = 6((x + 1)^2 - 1) - 10$$

Simplifying the Equation

To simplify the equation, we distribute the 6 to the terms inside the parentheses.

$$y = 6(x + 1)^2 - 6 - 10$$

Combining like terms, we get:

$$y = 6(x + 1)^2 - 16$$

Comparing with the Options

Now, let's compare our rewritten equation with the options provided.

• Option A: $y = 6(x+1)^2 - 11$
• Option B: $y = 6(x+1)^2 - 10$
• Option C: $y = 6(x+1)^2 - 4$
• Option D: $y = 6(x+1)^2 - 16$

Our rewritten equation matches with Option D: $y = 6(x+1)^2 - 16$.

Conclusion

In conclusion, the equation $y = 6x^2 + 12x - 10$ is rewritten in vertex form as $y = 6(x+1)^2 - 16$. This form is useful for identifying the vertex and the direction of the parabola's opening.

Key Takeaways

• Vertex form is a way of expressing quadratic equations in the form of $y = a(x-h)^2 + k$.
• Completing the square is a method used to rewrite quadratic equations in vertex form.
• The vertex form is useful for identifying the vertex and the direction of the parabola's opening.

Practice Problems

Try rewriting the following quadratic equations in vertex form:

• $y = 2x^2 + 4x - 3$
• $y = 3x^2 - 6x + 2$
• $y = 4x^2 + 8x - 1$

Additional Resources

For more information on quadratic equations and vertex form, check out the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Vertex Form of a Quadratic Equation
• Purplemath: Vertex Form of a Quadratic Equation
  Quadratic Equations in Vertex Form: Q&A =============================================

Frequently Asked Questions

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 coordinates of the vertex of the parabola.

Q: Why is vertex form useful?

A: Vertex form is useful for identifying the vertex and the direction of the parabola's opening. It also makes it easier to graph the parabola and find the x-intercepts.

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 factoring out the coefficient of $x^2$, adding and subtracting the square of half the coefficient of $x$ inside the parentheses, and then simplifying the equation.

Q: What is completing the square?

A: Completing the square is a method used to rewrite quadratic equations in vertex form. It involves adding and subtracting a constant term inside the parentheses to create a perfect square trinomial.

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 off the coordinates of the vertex from the equation. The vertex is represented by the point $(h, k)$.

Q: Can I use vertex form to find the x-intercepts of a parabola?

A: Yes, you can use vertex form to find the x-intercepts of a parabola. To do this, you need to set the equation equal to zero and solve for $x$.

Q: What are some common mistakes to avoid when rewriting quadratic equations in vertex form?

A: Some common mistakes to avoid when rewriting quadratic equations in vertex form include:

• Not factoring out the coefficient of $x^2$
• Not adding and subtracting the square of half the coefficient of $x$ inside the parentheses
• Not simplifying the equation correctly
• Not reading off the coordinates of the vertex correctly

Q: How do I graph a parabola in vertex form?

A: To graph a parabola in vertex form, you can use the coordinates of the vertex to find the x-intercepts and the y-intercept. You can then use this information to plot the parabola on a coordinate plane.

Q: Can I use vertex form to find the equation of a parabola given its vertex and a point on the parabola?

A: Yes, you can use vertex form to find the equation of a parabola given its vertex and a point on the parabola. To do this, you need to use the coordinates of the vertex and the point to find the values of $a$, $h$, and $k$ in the equation.

Additional Resources

For more information on quadratic equations and vertex form, check out the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Vertex Form of a Quadratic Equation
• Purplemath: Vertex Form of a Quadratic Equation

Practice Problems

Try rewriting the following quadratic equations in vertex form:

• $y = 2x^2 + 4x - 3$
• $y = 3x^2 - 6x + 2$
• $y = 4x^2 + 8x - 1$

Try finding the vertex and x-intercepts of the following parabolas in vertex form:

• $y = 2(x-2)^2 + 1$
• $y = 3(x+1)^2 - 2$
• $y = 4(x-1)^2 + 3$

Conclusion

In conclusion, vertex form is a useful way of expressing quadratic equations that makes it easier to identify the vertex and the direction of the parabola's opening. By completing the square and simplifying the equation, you can rewrite a quadratic equation in vertex form and use it to find the vertex and x-intercepts of the parabola.