Vertex (Quadratic Functions)

Definition

The vertex of a quadratic function is the point on the graph where the function reaches its maximum or minimum value. It is the highest or lowest point on the parabola, depending on the direction in which the parabola opens.

Key points about the vertex:

• If the parabola opens upward (i.e., when the coefficient a is positive), the vertex represents the minimum point of the graph.

• If the parabola opens downward (i.e., when the coefficient a is negative), the vertex represents the maximum point of the graph.

• The vertex is located on the axis of symmetry, which is a vertical line that divides the parabola into two symmetric halves.

How to find the vertex:

For a quadratic equation in standard form y = ax^2 + bx + c, the x-coordinate of the vertex can be found using the formula:

Once you find the x-coordinate, you can substitute it back into the quadratic equation to find the corresponding y-coordinate.

Example:

For the quadratic equation y = 2x^2 -4x + 1, the x-coordinate of the vertex can be found as:

Now, substitute x=1 back into the equation to find the y-coordinate:

y = 2(1)^2 - 4(1) + 1 = 2 - 4 + 1 = -1

So, the vertex is at (1, -1).

In summary, the vertex is the point where the quadratic function reaches its maximum or minimum value, and it is located on the axis of symmetry of the parabola.

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