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What information does the vertex form of a quadratic equation provide?

Published in Quadratic Equations 2 mins read

The vertex form of a quadratic equation provides the vertex coordinates (h, k) and indicates whether the parabola opens upwards or downwards.

Here's a breakdown of what the vertex form tells you:

Understanding Vertex Form

The vertex form of a quadratic equation is expressed as:

f(x) = a(x - h)^2 + k

Where:

  • f(x) represents the value of the quadratic function at a given x.
  • a determines the direction and stretch of the parabola.
  • (h, k) represents the coordinates of the vertex.

Key Information Provided by Vertex Form

  • Vertex Coordinates (h, k): The most crucial information is the vertex itself. The vertex is the minimum or maximum point of the parabola. Specifically, the x-coordinate of the vertex is h, and the y-coordinate is k. Remember that in the equation (x - h), the x-coordinate of the vertex is h, not -h.

  • Direction of Opening (Determined by 'a'):

    • If a > 0 (a is positive), the parabola opens upwards. This means the vertex is the minimum point of the parabola.
    • If a < 0 (a is negative), the parabola opens downwards. This means the vertex is the maximum point of the parabola.

  • Vertical Stretch or Compression (Determined by 'a'):

    • If |a| > 1, the parabola is vertically stretched (narrower).
    • If 0 < |a| < 1, the parabola is vertically compressed (wider).
    • If |a| = 1, the parabola has a standard width.

Example

Consider the equation:

f(x) = 2(x - 3)^2 + 4

From this equation, we can directly determine:

  • The vertex is at (3, 4).
  • Since a = 2 (positive), the parabola opens upwards, and the vertex is a minimum.
  • Since |a| = 2 > 1, the parabola is vertically stretched (narrower than the standard parabola).

Converting from Standard Form to Vertex Form

The standard form of a quadratic equation is:

f(x) = ax^2 + bx + c

You can convert from standard form to vertex form by completing the square. This allows you to easily identify the vertex.

Table Summarizing Information

Element Information Provided
h x-coordinate of the vertex
k y-coordinate of the vertex
a Direction and vertical stretch/compression of the parabola

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