The lowest point of a parabola, also known as its vertex, is a crucial feature that indicates either the minimum or maximum value of a quadratic function. For a parabola described by the standard quadratic equation y = ax² + bx + c, its lowest point (or highest point, if it opens downwards) can be precisely determined using specific formulas for its x and y coordinates.
Understanding the Standard Quadratic Form
A parabola is the graphical representation of a quadratic equation in the form:
y = ax² + bx + c
Here:
- a: Determines the direction the parabola opens and its vertical stretch or compression.
- If a > 0, the parabola opens upwards, and the vertex is the lowest point (a minimum).
- If a < 0, the parabola opens downwards, and the vertex is the highest point (a maximum).
- b: Affects the position of the axis of symmetry and the vertex.
- c: Represents the y-intercept of the parabola (where the curve crosses the y-axis).
Formulas for the Vertex (Lowest Point)
To find the exact coordinates (h, k) of the vertex, where h is the x-coordinate and k is the y-coordinate, you use the following formulas:
X-coordinate of the Vertex (Horizontal Position)
The formula for the x-coordinate (h) of the vertex is:
h = -b / 2a
This formula gives you the horizontal position of the lowest (or highest) point of the parabola.
Example from Reference:
For an equation where a = 2 and b = -6, the horizontal coordinate h would be calculated as:
h = -(-6) / (2 * 2)
h = 6 / 4
h = 3/2
Y-coordinate of the Vertex (Minimum Value)
Once you have the x-coordinate (h), you can find the y-coordinate (k) of the vertex. This y-coordinate represents the minimum value of the parabola if it opens upwards. The formula for the y-coordinate (k) is:
k = c - b² / 4a
Alternatively, you can substitute the calculated h value back into the original quadratic equation y = ah² + bh + c to find k.
Summary of Vertex Formulas
Here's a quick reference for finding the coordinates of the lowest point (vertex) of a parabola in the form y = ax² + bx + c:
| Coordinate | Formula | Description |
|---|---|---|
| X | h = -b / 2a |
Horizontal position of the vertex |
| Y | k = c - b² / 4a |
Vertical position (minimum/maximum value) of the vertex |
Practical Insights and Tips
- Identifying the Lowest Point: A parabola only has a "lowest point" (a minimum) if its a value is positive (a > 0). If a is negative (a < 0), the parabola opens downwards and has a "highest point" (a maximum) instead. The formulas for the vertex remain the same for both cases; they simply identify the extremum (either minimum or maximum).
- Vertex as the Axis of Symmetry: The vertical line x = h (where h is the x-coordinate of the vertex) is the axis of symmetry for the parabola, meaning the parabola is a mirror image on either side of this line.
- Applications: Understanding the lowest or highest point of a parabola is crucial in various fields, including physics (e.g., trajectory of a projectile), engineering (e.g., bridge design), and economics (e.g., optimizing profit or cost functions).
