Conic Sections Univerity of South Florida：圆锥截面南佛罗里达州大学

1、colleen beaudoinfebruary, 2009review: the geometric definition relies on a cone and a plane intersecting italgebraic definition: a set of points in the plane such that the difference of the distances from two fixed points, called foci, remains constant.xyfrom each point in the plane, the difference

2、of the distances to the foci is a constant. example:f1f2d1d2focipoint a: d1-d2 = c point b: d1-d2 = c bad1d2xyf1f2focicentertransverse axisconjugate axisverticesalgebraic definition of a hyperbola: a set of points in the plane such that the difference of the distances from two fixed points, called f

3、oci, remains constant.how is the definition similar to that of an ellipse?how is it different?qboth variables are squared. qequation: qcompare the equations of ellipses and hyperbolas.qwhat makes the hyperbola different from the parabola?qwhat makes the hyperbola different from a circle?22222222( -

4、)( - )( - )( - )1 or x hy ky kx hababprocedure to graph:1. put in standard form (above): x squared term - y squared term = 12. determine if the hyperbola is opening vertically or horizontally. (if x is first, its horizontal. if y is first, its vertical.)3. plot the center (h,k)4. plot the endpoints

5、of the horizontal axis by moving “a” units left and right from the center.22222222( - )( - )( - )( - )1 or x hy ky kx habbato graph:5. plot the endpoints of the vertical axis by moving “b” units up and down from the center.note: the line segment that contains the vertices of the hyperbola is known a

6、s the transverse axis. the other axis is the conjugate axis.6. draw a rectangle such that each of the axis endpoints is the midpoint of a side.22222222( - )( - )( - )( - )1 or x hy ky kx habbato graph:7. sketch the diagonals of the rectangle and extend them outside of the rectangle. (these are the a

7、symptotes of the hyperbola.)8. draw each branch of the hyperbola be sure to go through the vertex of each (the endpoint of the transverse axis) and approach the asymptotes.22222222( - )( - )( - )( - )1 or x hy ky kx habbato graph:9. use the following formula to help locate the foci: c2 = a2 + b2 mov

8、e “c” units left and right form the center if the transverse axis is horizontal or move “c” units up and down form the center if the transverse axis is verticallabel the points f1 and f2 for the two foci.note: it is not necessary to plot the foci to graph the hyperbola, but it is common practice to

9、locate them. 22222222( - )( - )( - )( - )1 or x hy ky kx habbathe equation of each asymptote can be found by using the point-slope formula. use the center as “the point” and slope can be found by counting on the graph (from the point to the corner of the rectangle).or the following formulas can be u

10、sed:with horizontal transverse axis: bby = k + (x - h) and y = k - (x - h)aawith vertical transverse axis: aay = k + (x - h) and y = k - (x - h)bb1. put in standard form. done2. determine if the hyperbola is opening vertically or horizontally. vertically because “y” is first.3. identify the center.

11、(0,0)4. identify the endpoints of the horizontal axis. (6,0) and (-6,0)5. identify the endpoints of the vertical axis. (0,8) and (0,-8) which pair of endpoints are the vertices? (0,8) and (0,-8)2216436yx6. draw a rectangle such that each of the axis endpoints is the midpoint of a side.7. sketch the

12、asymptotes of the hyperbola.8. draw each branch of the hyperbola be sure to go through the vertex of each (the endpoint of the transverse axis) and approach the asymptotes.2216436yx9. locate the foci. (0,10) and (0,-10) 10. find the equations of the asymptotes. 2216436yx44 and 33yxyx xytransverse ax

13、isconjugate axiscenterasymptotes2216436yx 22(3)(2)13616xy1. put in standard form. done2. determine if the hyperbola is opening vertically or horizontally. horizontally because “x” is first.3. identify the center. (3,-2)4. identify the endpoints of the horizontal axis. (-3,-2) and (9,-2)5. identify t

14、he endpoints of the vertical axis. (3,2) and (3,-6) which pair of endpoints are the vertices? (-3,-2) and (9,-2)6. draw a rectangle such that each of the axis endpoints is the midpoint of a side.7. sketch the asymptotes of the hyperbola.8. draw each branch of the hyperbola be sure to go through the

15、vertex of each (the endpoint of the transverse axis) and approach the asymptotes.22(3)(2)13616xy9. locate the foci. (3+213,-2) and (3-213,-2)10. find the equations of the asymptotes. 44 and 33yxyx22(3)(2)13616xy22(3)(2)13616xy xy22464yx2216416yx1. put in standard form. 2216416yx xywrite the equation for a hyperbola with x-intercepts at 5 and -5 and foci (6,0) and (-6,0).1)how can you tell if the graph of an equation will be a line, parabola,