How to Find the Vertex of a Parabolic Function

The vertex of a parabolic function is a crucial point that can provide valuable information about the graph of the function. It is the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards. Finding the vertex allows us to determine key features such as the maximum or minimum value of the function.

Understanding the Vertex Form of a Quadratic Function

To find the vertex of a parabolic function in the form y = ax^2 + bx + c, we can use a process called completing the square to convert it into vertex form, y = a(x-h)^2 + k. In this form, (h, k) represents the coordinates of the vertex.

Step-by-Step Guide to Finding the Vertex

1. Identify coefficients: Start by identifying the values of a, b, and c in your quadratic equation, y = ax^2 + bx + c.

2. Calculate h: Use the formula h = -b/(2a) to find the x-coordinate of the vertex.

3. Substitute h back into equation: Substitute the value of h back into your equation to find k, which represents the y-coordinate of the vertex.

4. Determine Vertex: Once you have found both h and k, you have successfully determined the coordinates of the vertex, which is represented as (h, k).

5. Interpretation: Depending on whether your parabola opens upwards or downwards, you can determine if it is a maximum or minimum point respectively.

Example: Finding Vertex of a Parabolic Function

Let’s consider an example: Find the vertex of the function given by:
y = 2x^2 – 8x + 6

1. Identify coefficients: In this case, a = 2, b = -8, and c = 6.

2. Calculate h: Using formula,
h = -(-8) / (2*2)
h = 8 / 4
h = 2

3. Substitute h back into equation:
y = 2(x-2)^2 + k

4. Determine k:
Plug in x=2 into original equation,
y = 2(2)^2 – 8(2) + 6
y = 8 -16 +6
y= -2

5. Interpretation:
The vertex is at (2,-2), representing a minimum point on this upward opening parabola.

By following these steps, you can effectively find and interpret vertices for various parabolic functions, allowing for deeper insights into their graphs and characteristics.