Quadratic expressions and parabolas
What does ax squared plus bx plus c look like, and why does it curve?
Where you are 4 inside
What this area is about
A linear expression graphs as a straight line. Add an x^2 and the line bends into a parabola: a U shape that turns around exactly once and is the same on both sides. This area is about what makes an expression quadratic and how to read the parabola it draws.
How these fit together
The definition. What makes an expression quadratic: degree two, so the variable is squared and no higher, and a \neq 0.
The shape. The shape of a parabola explains why squaring gives a symmetric curve, and why a > 0 opens up while a < 0 opens down.
Its landmarks. The vertex and the axis of symmetry are taken together. The axis is the line \displaystyle x = -\frac{b}{2a} that the curve folds onto, and the vertex sits on that line as the highest or lowest point.
The procedure. Graphing from standard form puts the landmarks together: axis, vertex, intercepts, then sketch.
A path through it
Definition, then shape, then the vertex and its axis, then graphing. Each one uses the one before.
With the picture in hand, the real solutions of a quadratic equation are where the parabola meets the x-axis, which is what roots, zeros and x-intercepts makes precise. Factoring and square roots find those points, and this area is the ground for quadratic functions.