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
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.