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urp:graphing
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urp:graphing [2021-10-21] (current) nerf_herder |
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| ====Graphing curves and circles==== | ====Graphing curves and circles==== | ||
| - | vertex of a parabola: for y = ax^2 + bx + c, then x = -b/2a | + | === Circles === |
| - | + | ||
| - | standard form of a parabola: | + | |
| - | (x-h)² = 4p(y-k) => if p>0, opens up, p<0 opens down | + | |
| - | (y-k)² = 4p(x-h) => if p>0, opens to right, p<0 opens to left | + | |
| - | where point (h,k) is the vertex, and | + | |
| - | p = minimum distance between parabola and vertex (is on axis of symmetry, | + | |
| - | which is perpendicular to the directrix) | + | |
| - | LR (latus rectum line) is line parallel to directrix going thru focus (if you know focus, | + | |
| - | easy to find LR points and vertex, then draw the function). Length of LR is |4p| | + | |
| - | + | ||
| Standard form of a circle: | Standard form of a circle: | ||
| (x-h)² + (y-k)² = r² | (x-h)² + (y-k)² = r² | ||
| Line 18: | Line 8: | ||
| Can go from general form to standard form by completing the square | Can go from general form to standard form by completing the square | ||
| + | ===Ellipse=== | ||
| Ellipse - sum of distance from two foci is a constant | Ellipse - sum of distance from two foci is a constant | ||
| points on the ellipse follow the formula: | points on the ellipse follow the formula: | ||
| Line 29: | Line 20: | ||
| (x,y) = a point on the ellipse | (x,y) = a point on the ellipse | ||
| + | ===Parabolas=== | ||
| + | Any point on a parabola is equidistant from the focus and the directrix | ||
| + | |||
| + | vertex of a parabola in quadratic form: for y = ax^2 + bx + c, then x = -b/2a | ||
| + | {{ :urp:parabola.jpg?400|}} | ||
| + | |||
| + | standard form of a parabola: | ||
| + | (x-h)² = 4p(y-k) => if p>0, opens up, p<0 opens down | ||
| + | (y-k)² = 4p(x-h) => if p>0, opens to right, p<0 opens to left | ||
| + | where point (h,k) is the vertex, and | ||
| + | p (sometimes called a instead) = minimum distance between | ||
| + | parabola and vertex (is on axis of symmetry, which is | ||
| + | perpendicular to the directrix) | ||
| + | this can be rewritten as: | ||
| + | y = (1/4p)(x-h)² + k for an up/down parabola | ||
| + | x = (1/4p)(y-k)² + h for a left/right parabola | ||
| + | LR (latus rectum line) is line parallel to directrix going | ||
| + | thru focus (if you know focus, easy to find LR points | ||
| + | and vertex, then draw the function). Length of LR is |4p| | ||
| + | |||
| + | ===Hyperbola=== | ||
| Hyperbola - difference of distance from two foci is a constant | Hyperbola - difference of distance from two foci is a constant | ||
| - | Back to [[geometry]] or [[math]] page. | + | If there are foci F and G, point P then distances |PF - PG| = C |
| + | |||
| + | Opens left/right: | ||
| + | (x-h)²/a² - (y-k)²/b² = 1 | ||
| + | center at (h, k) | ||
| + | vertices //a// units left/right of center | ||
| + | asymptotes pass through center with slope ± b/a. | ||
| + | |||
| + | Opens up/down: | ||
| + | (y-k)²/b² - (x-h)²/a² = 1 | ||
| + | center at (h, k) | ||
| + | vertices //b// units up/down of center | ||
| + | asymptotes pass through center with slope ± b/a. | ||
| + | |||
| + | Back to [[geometry]], [[algebra|algebra/pre-calc]] or [[math]] page. | ||
urp/graphing.1634102474.txt.gz · Last modified: 2021-10-13 by nerf_herder
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