Quadratic Functions

Where this is used

Raise the price and you lose customers; drop it and you lose profit. The "just right" point in between is the vertex of a quadratic function — something economists, engineers, and physicists reach for every day.

Start gently

A quadratic is fixed by two things: where the vertex sits and how widely it opens. Completing the square is just a rewrite that lets you read the vertex straight off the formula.

Key points

What a quadratic function is

A function written as y = ax² + bx + c. If a is positive the graph opens upward (a U shape); if a is negative it opens downward (an upside-down U). The turning point, called the vertex, is where the function reaches its lowest or highest value.

Vertex form

Rewrite the function as y = a(x - p)² + q and you can read the vertex straight off: it is (p, q). Getting it into that shape is called completing the square.

The x-coordinate of the vertex

For y = ax² + bx + c, the vertex sits at x = -b/(2a). That is exactly where the maximum or minimum happens.

Using it in business

Suppose a product priced at ¥x brings in revenue y = -2x² + 1000x. The vertex is at x = 250, so ¥250 is the price that earns the most. Real pricing strategy starts from models like this one.

The discriminant

How many times the graph crosses the x-axis (the solutions of y = 0) is decided by the discriminant D = b² - 4ac. If D > 0 there are two crossings, if D = 0 there is one, and if D < 0 there are none.

See it drawn

y = x² − 4x + 5 = (x − 2)² + 1

Completing the square gives (x − 2)² + 1. The vertex sits at x = 2, where the bracket becomes 0.

Jobs that use this

Data Scientist$120k

Product Pricing Strategy Manager$130k

Game Physics Engineer$110k

Quadratic Functions

Start gently

Key points

See it drawn

Jobs that use this

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