Chapter 4
Quadratic Functions and Equations
Throw a ball straight up. It climbs, slows, stops for an instant, then falls back. No straight line does that; a line that is rising keeps rising forever. The same turning shape appears in the arc of water from a hose, and in the area of a rectangle whose perimeter is fixed. What these share is a single squared term in the rule, and that squared term is what makes a relationship quadratic. One term is enough to bend a graph. What else does it decide?
What You'll Explore
6 lessons.
- Quadratic Functions and Parabolas
Are all parabolas the same curve underneath, just moved and stretched? And why does one quadratic seem to come in several different written forms? This opening lesson pins down what makes a function quadratic, and looks at what a change of form buys you when you want to read something off the rule.
- Solving Quadratics by Factoring
Rewriting a quadratic as a product of two simpler pieces does not change the equation at all, so why does it suddenly hand you the solutions? One fact about ordinary numbers is doing the real work here, and tracking it down also reveals where this method quietly runs out.
- Completing the Square
Suppose no whole numbers will factor a quadratic, yet its graph plainly crosses the axis twice. Something must still be there to find. This lesson builds a method that never has to guess, reshaping any quadratic at all around a single perfect square, and its name comes from a real picture: a square with one corner missing.
- The Quadratic Formula and the Discriminant
Running the same method again for every new set of numbers is wasteful. What if you ran it once, keeping the coefficients as letters, and kept the answer forever? That single computation is the quadratic formula, derived rather than handed down, and buried inside it is one number worth knowing before you solve.
- Quadratic Inequalities
Most real questions are not about the one instant a quantity hits zero. A business turns a profit across a range of prices, not at a single price, and a fenced field can beat a target area for a whole range of widths. Asking where a quadratic is positive or negative changes what an answer even looks like.
- Sum and Product of Roots
Every method so far has pushed in one direction, from the coefficients toward the roots. What happens if you run the map backwards and expand a factored quadratic out again? The chapter closes by asking what the coefficients were quietly saying about the two roots all along.