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Quadratic Concepts

I will try to write this in plain English, not technical “math-speak”.

A quadratic equation is an equation that can be written in this form:

y = ax² + bx + c (where a, b, and c are numbers)

For example, y = 4x² + 3x – 6.

There are a few forms we (us mathematicians) prefer to see:

General Form: y = ax² + bx + c

or

Standard Form: y = a(x – h)² + k

1.  Key Concepts:

  • Shape: If “a” is a positive number, the parabola (graph shape) makes a smiley face.  If “a” is a negative number, we have a frowny face.
  • Maximum or Minimum?:  A smiley face has a minimum (low point).  A frowny face has a maximum (high point).  The maximum or minimum is called the vertex.  A vertex is a single point (x, y).

2. How to find the vertex:

  • In general form y = ax² + bx + c, the x -coordinate of the vertex is (-b/2a).

Example:  y = -4x + 8x – 7.  The x-coordinate = -8 / (2 * -4) = -8 / -8 = 1.  Now, to find the y-coordinate of the vertex, we plug in x = 1 to the original equation:  y = -4(1) + 8(1) – 7 = -3.  The vertex is (1, -3).  Since “a” is negative, this is a frowny face parabola which means (1, -3) is a maximum.

  • In standard form y = a(x – h)² + k, the vertex is staring you in th face.  It is the point (h, k).

Example 1: y = 7(x – 9)² + 89.  The vertex is (9, 89).  This vertex is a minimum.

Example 2: y = -12(x + 4)² – 11.  The vertex is (-4, -11).  This vertex is a maximum.


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