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What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?

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The family of graphs of x^n + y^n =1 (for even n) includes the circle. Why do the graphs look more and more square as n increases?

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What Do Functions Do for Tiny X?

Looking at small values of functions. Motivating the existence of the Taylor expansion.

Loch Ness

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Remember that the function $f(x) = |x|$ is defined as:

$$\eqalign{ y &= x \ {\rm for } \ x\geq 0 \cr &= -x \ {\rm for } \ x < 0 .}$$

Split up the domain of the function.