 ## Function and Relation Library

Conic Sections A traditional view of the conics is included.

Pick a point in space to use as a pivot point and vertex. Pass a line through the point and rotate it tracing two cones. Each cone is called a nape. Next take a plane and think of the ways the place may cut the and cones. Lines, parabolas, circles, ellipses, hyperbolas, and points are all conic sections, or curves which are formed when one or both napes are cut by the plane.

For our purposes, conics must be written as the union of two functions. The traditional conic section equations, must be rewritten as expressions for y. In each case, one manipulation of the equation looks like "y² = ..." or "(y - k)² = ..." When the square root of each side is taken, two seperated functions for y result, one is negative one is positive.

The remainer of the page is meant only as examples, not as an exhaustive list of important points.

A PARABOLA The Squaring and Square Root Functions from Function and Relation Library.

The Quadratic, often written as y = ax² + bx + c or y = a(x - h)² + k, is a parabola.  Visit the Interactive Sketch Pad MaterialOn Parabolas, Families of Quadratics parabola   -- Multi page: 2 parabolas: 1 in general form, 1 in standard forms,     * both generated by equations defined by parameters.     * WITH SHOW/HIDE ACTION BUTTON     * graphs of 2 parabolas: 1 in general form, 1 in standard forms,     * 1 parabola, set a, then generat parabola w/2 points.     * 1 parabola controlled by 3 points, used for determinant analysis     * 1 vertical axis of symmetry, 1 horizontal "axis of symmetry TracePoly   -- drag point f(x) to state and show new ( x, f(x))

A CIRCLE The circle x² + y² = r² having a radius of r and centered at (0,0) may be graphed as the union of the two functions

y = (r² - x²) and y = - (r² - x²).  Visit the Interactive Sketch Pad Material On Circles GraphGivenCircle   -- Graph a circle by graphing + and - functions defined by parameters.     * Also states equation of circle.

The circle x² + y² = 3² having a radius of 3 and centered at (0,0) may be graphed as the union of the two functions

y = (3² - x²) and y = - (3² - x²).

The circle (x - h)² + (y - k)² = r² having a radius of r and a center at (h,k) must be graphed as the union of functions:  Examples follow. Circles with:center at (1,3) and radius of 4 center at (-2,-1) and radius of 1 center at (4,-4) and radius of 2

AN ELLIPSE The standard equation for the ellipse, also requires two function equations:  The center is (h,k) and 2a and 2b are the lengths of the major and minor axes.

A HYPERBOLA The standard equation for the hyperbola is like that of the ellipse. It requires two functions.  At left, asymptotes are graphed as well as the hyperbola.

y=f(x) Hyperbola or x=f(y) Hyperbola on SketchPad

This page is from Exploring Functions Throught the Use of Manipulatives (ISBN: 0-9623593-3-5).

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