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 gives the two moments the ball is at ground level, and nothing about the arc between them. The motion has a shape, though, and a picture reveals it. The same idea pins down a circle too, and turns a plain equation into a question of where a curve sits above zero, below it, or reaches its very best. So what can a picture tell you that numbers alone cannot?
What You'll Explore
4 lessons.
Some lessons are marked Advanced because they go beyond core Algebra I. Skip them or explore them if you are curious.
- 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, so their equation looks nothing like a parabola's, yet it still comes from one condition every point on the circle shares. What is that condition? Here you turn it into an equation, then read a circle's position and size back out of it.
- Quadratic Inequalities Advanced. This lesson goes beyond core Algebra I. You can skip it.
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.