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The vertex sits at x = −b/2a, which is why it slides sideways when you move b alone. The sign of a decides which way the curve opens, and its size decides how tight it is.
Change the coefficients of a quadratic, drag the plot to pan, and zoom with the buttons.
A quadratic on paper is a fact to memorise. A quadratic whose coefficients you can drag is something you build intuition for: you find out what a does by moving it. That matters most in math, because nearly everything later rests on whether the early material became intuition or stayed a formula.
Arraste o ponto ao longo da curva.
The slope at the point is the acceleration; the strips under the curve add up to the distance; and the distance curve's slope is the speed you started with. Three tabs, one fact.
For calculus, probe one curve: the slope at a point is a rate, the strips under it add up to a total, and the slope of that total is the curve you started with.
Zoom in far enough on any smooth curve and it becomes a straight line. That line is the tangent, and its slope is the derivative at that point.
Zoom in on any smooth curve and it straightens into its tangent. The slope of that line is the derivative, shown before it is defined.
Every height on the lower curve is the slope of the upper one at the same x. Where the top curve goes flat, the bottom one crosses zero.
Move x along a curve and the tangent's slope is plotted underneath, leaving a trail. The trail is the derivative, and its zeros sit under the flat spots.
Cut a square from each corner of a 12 × 12 sheet and fold up the sides. The volume peaks where its graph goes flat: at x = 2 the slope is zero and the box holds 128.
The classic optimisation problem as an object: cut the corners, fold the box, and watch the volume peak exactly where its slope is zero.
Every term you add makes the polynomial match sin x across a wider stretch. The shaded gap is everything it still gets wrong.
sin x rebuilt one term at a time. Each term of the series buys a wider stretch of the real curve, and the shaded gap is what it still gets wrong.
The sine of an angle is just the height of a point on the unit circle. Carry that height across as the angle grows and the wave draws itself.
Where sine comes from: a point goes round the unit circle, its height is carried across, and the wave draws itself. Cosine is the same point's shadow.
Each circle spins at an odd multiple of the first, smaller each time. Add enough of them and their sum draws a square wave, corners and all.
A square wave built from circles. Each one spins at an odd multiple of the first; chain enough of them and the last tip draws corners, overshoot and all.
You describe the goal in your own words and the plan asks questions back until it is one you would follow. Goals usually look like these.
There is no fixed syllabus. The plan is built around your goal and drops what it does not need. These are the topics it draws from.
Each one has its own lesson block to try.
See how it works and what it costs.