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Conic sections

Circles, ellipses, parabolas and hyperbolas as equations and as surfaces sliced.

Slice a cone and you get a circle, an ellipse, a parabola or a hyperbola; algebraically they are the second-degree equations in x and y. Completing the square in each variable reveals the centre and shape. The 3D view draws z = the expression, whose level curve z = 0 is the conic.

Àwọn ààtò àwọn ìṣàmúlò-ètò · x^2 + y^2 - 25

Algebra · Expressions

Analyse x^2 + y^2 - 25

x^{2} + y^{2} - 25

Àwọn ìṣàmúlò-ètò

x^{2} + y^{2} - 25
An expression in x, y. Here is what it does.

Fi àwọn àgbèwọlé hàn

x^{2} + y^{2} - 25

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Bii o ṣe le: Conic sections

An expression in x, y. Here is what it does.

Àwọn Àtòjọ-ẹ̀yàn

What is a function, really?

A rule that assigns exactly one output to each input. The vertical-line test on a graph is the same idea: no input may have two outputs.

Why do we need complex numbers?

Because x² + 1 = 0 has no real solution, and allowing one new number i with i² = −1 makes every polynomial equation solvable. They then turn out to describe rotation, waves and alternating current more naturally than real numbers do.

Diẹ̀ nínú Precalculus

Complex numbers Polynomial functions Rational functions Sequences and series The binomial theorem Vectors Exponential and logarithmic functions Polynomial division and the remainder theorem Parametric equations and polar coordinates