Transcription of Implicit Functions - Dartmouth College
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Implicit Functions - Dartmouth College
Implicit FunctionsDefining Implicit FunctionsUp until now in this course, we have only talked about Functions , which assign to every real numberxintheir domain exactly one real numberf(x). The graphs of a functionf(x) is the set of all points (x, y) suchthaty=f(x), and we usually visually the graph of a function as a curve for which every vertical line crossesthat curve at most once. There are other curves that we can draw on thexy-plane which do not pass thevertical line test. One such curve is the circle of radius 1 centered at the origin. We can describe this circlewith the relationx2+y2= 1,that is, the circle of radius 1 centered at the origin is the set of all points (x, y) such thatx2+y2= 1. Nowconsider one point on this circle, the point (0,1). You may notice that if we remove some of the circle (forexample, the lower half of the circle), the remaining curve is the graph of a function.
The curve of a conic section is a parabola when b2 ¡ 4ac = 0. Here we allow for the parabola to be open in any direction, including horizontally. So, for example, the conic section x¡y2 +3x¡2 = 0 gives us a parabola which is open to the right. † A hyperbola, roughly speaking, is a curve which consists of two disconnected parabola-like ...
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