Chapter 10
Graphing Quadratics and Inequalities
Throw a ball straight up and it does not rise forever. It slows, turns, and falls back, and the way up mirrors the way down. Solving a quadratic tells you the two moments the ball is at ground level, and nothing at all about the arc in between. Those answers are numbers, but the motion has a shape. Drawing a quadratic relationship, rather than only solving it, puts that shape in front of you. So what can a picture tell you that a pair of numbers cannot?
What You'll Explore
4 lessons.
- Parabolas
Plot a handful of quadratics and a family resemblance shows up: every one bends into a smooth curve, yet some open upward, some downward, some narrow, some wide. What in the equation settles which, and what fixes the point where the curve turns around? This lesson ties a parabola's shape and position to the numbers in its equation.
- The Equation of a Circle
Circles close on themselves instead of opening outward, yet they live in the same plane and can be pinned down by the same kind of equation. What single condition does every point on a circle share? Here you turn that condition into an equation, then read a circle's position and size back out of it.
- Quadratic Inequalities
Equations ask when a quadratic equals zero. A bridge clearance or a profit target asks something looser: for which inputs does the expression stay above zero, or drop below it? Instead of two exact answers you want whole stretches of the number line, and this lesson looks at how the curve decides which stretches those are.
- Quadratic Optimization
Give a farmer a fixed length of fence and some rectangles enclose more ground than others, while one encloses the most. Ticket prices and running costs hide a best case in the same way. When a quadratic describes the quantity, this lesson looks for that best case, and separates the best value from the choice that produces it.