大学课件：parabolas.ppt

1、Colleen BeaudoinFor FCITGeometric definition:a cone has a planeintersecting it,crossing itsvertical axisAlgebraic definition:All points that are equidistant from a given line(the directrix)and a fixed point not on the directrix(the focus)yAny point on the parabola is equidistant to the focus and the

2、 directrix.Example:Point A:d1=d1Point B:d2=d2ABFocusVertexDirectrixAxis of SymmetryxyOne variable is squared and one is not.(How does this differ from linear equations?)There are many ways the equation of a parabola can be written.We will get the quadratic part(variable that is squared)on the left o

3、f the equal sign and the linear part(variable is to the first power)on the right of the equal sign.Equation:(x-h)2=c(y k)OR (y-k)2=c(x h)To graph:1.Put in standard form(above)squared term on left2.Decide which way the parabola opens.Look at the right side.If y:+c opens up If y:-c opens down If x:+c

4、opens right If x:-c opens leftTo graph:3.Plot the vertex(h,k)Note what happens to the signs.4.Plot the focus:move c from the vertex in the direction that the parabola opens.Mark with an f.5.Draw the directrix:c from the vertex in the opposite direction of the focus(Remember that the directrix is a l

5、ine.)To graph:6.Plot the endpoints of the latus rectum/focal chord(width at the focus).The width is the c at the focus.7.Sketch the parabola by going through the vertex and the endpoints of the latus rectum.(Be sure to extend the curve and put arrows.)8.Identify the axis of symmetry.(The line that g

6、oes through the vertex dividing the parabola in half.)To graph:1.Put in standard form squared term on left Done2.Decide which way the parabola opens.Look at the right side.If y:+c opens up If y:-c opens down If x:+c opens right If x:-c opens leftUp because y is on the right and 12 is positiveTo grap

7、h:3.Plot the vertex(h,k)Note what happens to the signs.(5,6)4.Plot the focus:move c from the vertex in the direction that the parabola opens.Mark with an f.(5,9):found by moving up 3 from the vertex5.Draw the directrix:c from the vertex in the opposite direction of the focus(Remember that the direct

8、rix is a line.)y=3:found by moving down 3 from the vertex To graph:6.Plot the endpoints of the latus rectum/focal chord(width at the focus).The width is the c at the focus.L.R.=12 with endpoints at(-1,9)&(11,9)7.Sketch the parabola by going through the vertex and the endpoints of the latus rectum.(B

9、e sure to extend the curve and put arrows.)8.Identify the axis of symmetry.(The line that goes through the vertex dividing the parabola in half.)x=5fTo graph:1.Put in standard form Done2.Decide which way the parabola opens.Left because x is on the right and 4 is negative3.Plot the vertex(h,k)(2,-3)4

10、.Plot the focus:(1,-3):found by moving left 1 from the vertex5.Draw the directrix:x=3:found by moving right 1 from the vertex To graph:6.Plot the endpoints of the latus rectum L.R.=4 with endpoints at(1,-1)&(1,-5)7.Sketch the parabola8.Identify the axis of symmetry.y=-3xyfWhats the first step?Put in

11、 standard form.y2 -4y+1=x y2 -4y+4=x-1+4 Complete the square.(y 2)2=x+3(y 2)2=1(x+3)Now you try graphing the parabola and labeling all the parts.Given the following information,write the equation of the parabola.Vertex is(0,0)and Focus is at(0,2)1)How can you tell the graph of an equation will be a parabola?2)Whats the standard form of a parabola?3)What are the steps for graphing a parabola?4)What are common errors people make when graphing parabolas?Conic SectionStandard Form of EquationParabolaCircleEllipseHyperbola