The coordinates of the vertex for a quadratic equation in the standard form y = ax² + bx + c are found using specific formulas derived from its coefficients a, b, and c.
The vertex of a parabola (the graph of a quadratic equation) is the point where it reaches its maximum or minimum value. This critical point's coordinates can be expressed entirely in terms of a, b, and c.
The Vertex Formula
For a quadratic equation y = ax² + bx + c, the coordinates of the vertex (h, k) are given by:
- x-coordinate (h): $-b / 2a$
- y-coordinate (k): $a(-b / 2a)^2 + b(-b / 2a) + c$ (which simplifies to $\frac{4ac - b^2}{4a}$)
This means the vertex can be written as: $\left( \frac{-b}{2a}, \frac{4ac - b^2}{4a} \right)$
Steps to Find the Vertex Coordinates
To determine the exact coordinates of the vertex in terms of a, b, and c, follow these steps:
- Identify Coefficients: Ensure your quadratic equation is in the standard form y = ax² + bx + c. Clearly identify the values of a, b, and c.
- Calculate the x-coordinate: The x-coordinate of the vertex is found using the formula:
- $x = \frac{-b}{2a}$
This formula directly provides the axis of symmetry for the parabola.
- $x = \frac{-b}{2a}$
- Calculate the y-coordinate: Substitute the calculated x-coordinate back into the original quadratic equation y = ax² + bx + c.
- $y = a\left(\frac{-b}{2a}\right)^2 + b\left(\frac{-b}{2a}\right) + c$
This expression, when algebraically simplified, becomes: - $y = \frac{b^2}{4a} - \frac{b^2}{2a} + c$
- $y = \frac{b^2 - 2b^2}{4a} + c$
- $y = \frac{-b^2}{4a} + c$
- $y = \frac{-b^2 + 4ac}{4a}$
So, the y-coordinate simplifies to $\frac{4ac - b^2}{4a}$.
- $y = a\left(\frac{-b}{2a}\right)^2 + b\left(\frac{-b}{2a}\right) + c$
Summary Table
Coordinate | Formula in terms of a, b, c |
---|---|
x | $\frac{-b}{2a}$ |
y | $\frac{4ac - b^2}{4a}$ |
Therefore, the coordinates of the vertex, expressed precisely in terms of a, b, and c, are $\left( \frac{-b}{2a}, \frac{4ac - b^2}{4a} \right)$.