1.12) Find The Vertex Of The Equation $y=3x^2-12x+6$.A. $(6,2$\] B. $(3,6$\] C. $(2,-6$\]

**Finding the Vertex of a Quadratic Equation: A Step-by-Step Guide**

What is the Vertex of a Quadratic Equation?

The vertex of a quadratic equation is the highest or lowest point on the graph of the equation. It is the point where the parabola changes direction, from opening upwards to downwards or vice versa. In this article, we will focus on finding the vertex of the quadratic equation $y=3x^2-12x+6$.

Why is Finding the Vertex Important?

Finding the vertex of a quadratic equation is important in various fields such as physics, engineering, and economics. It helps in understanding the behavior of the function, identifying the maximum or minimum value, and making predictions about the future behavior of the system.

How to Find the Vertex of a Quadratic Equation?

To find the vertex of a quadratic equation, we can use the following steps:

Step 1: Write the Quadratic Equation in Vertex Form

The vertex form of a quadratic equation is given by:

$y=a(x-h)^2+k$

where $(h,k)$ is the vertex of the parabola.

Step 2: Complete the Square

To convert the quadratic equation into vertex form, we need to complete the square. We can do this by adding and subtracting a constant term inside the parentheses.

$y=3x^2-12x+6$

$y=3(x^2-4x)+6$

$y=3(x^2-4x+4-4)+6$

$y=3(x-2)^2-12+6$

$y=3(x-2)^2-6$

Step 3: Identify the Vertex

Now that we have the quadratic equation in vertex form, we can identify the vertex by looking at the value of $h$ and $k$.

$y=3(x-2)^2-6$

The vertex is at $(h,k)=(2,-6)$.

Example Questions and Answers

Q1: What is the vertex of the quadratic equation $y=x^2-6x+5$?

A. $(3,4)$ B. $(4,3)$ C. $(3,-4)$

Answer: C. $(3,-4)$

Q2: Find the vertex of the quadratic equation $y=2x^2+4x-3$.

Answer: The vertex is at $(h,k)=(-1,-5)$.

Q3: What is the vertex of the quadratic equation $y=x^2-2x-3$?

A. $(1,2)$ B. $(2,1)$ C. $(1,-2)$

Answer: C. $(1,-2)$

Conclusion

Finding the vertex of a quadratic equation is an important concept in mathematics and has various applications in real-life situations. By following the steps outlined in this article, we can easily find the vertex of a quadratic equation and understand its behavior.

Frequently Asked Questions

Q1: What is the vertex form of a quadratic equation?

A. $y=ax^2+bx+c$ B. $y=a(x-h)^2+k$ C. $y=ax^2-bx+c$

Answer: B. $y=a(x-h)^2+k$

Q2: How do I find the vertex of a quadratic equation?

A. By completing the square B. By using the quadratic formula C. By graphing the equation

Answer: A. By completing the square

Q3: What is the significance of finding the vertex of a quadratic equation?

A. It helps in understanding the behavior of the function B. It helps in identifying the maximum or minimum value C. It helps in making predictions about the future behavior of the system

Answer: C. It helps in making predictions about the future behavior of the system