210.1 ParabolasA parabola is the set of all points (x,y) that are equidistant from a fixed line (directrix) and a fixed point (focus) not on the line.Focus (h, k + p)Vertex (h,k)Directrix y = k - p
3Standard Equation of a Parabola (x - h)2 = 4p(y - k) Vertical axisOpens up (p is +) or down (p is -)(y - k)2 = 4p(x - h) Horizontal axisOpens right (p is +) or left (p is -)p is the distance from the center to the focuspoint.
4left Ex. Find the vertex, focus, and directrix of the parabola and sketch its graph. y2 + 4y + 8x - 12 = 0Now complete the square.y2 + 4y = -8x + 12y2 + 4y + 4 = -8x(y + 2)2 = -8x + 16Write down the vertexand plot it. Then find p.(y + 2)2 = -8(x - 2)4p = -8p = -2What does the negative pmean?left
6Right, since the axis is vertical, we will be using Ex. Find the standard form of the equation of theparabola with vertex (2,1) and focus (2,4).First, plot the two points.Which equation will we be using? Vert. or Horz. axisRight, since the axis is vertical, we will be using(x - h)2 = 4p(y - k)What is p?p = 3Now write down the equation.(x - 2)2 = 12(y - 1)
8EllipsesCenter point(h,k)Focus pointabFFcVVaMinor axisMajor axisAn ellipse is the set of all points (x,y), the sum of whose distances from two distinct points (foci) is constant.a2 = b2 + c2
9Standard Equation of an Ellipse Horz. Major axisVert. Major axis(h,k) is the center point.The foci lie on the major axis, c units from the center.c is found by c2 = a2 - b2Major axis has length 2a and minor axis has length 2b.
10Sketch and find the Vertices, Foci, and Center point. x2 + 4y2 + 6x - 8y + 9 = 0First, write the equation in standard form.(x2 + 6x ) + 4(y2 - 2y ) = -9(x2 + 6x + 9) + 4(y2 - 2y + 1) =(x + 3)2 + 4(y - 1)2 = 4C (-3,1)V (-1,1) (-5,1)
12Eccentricity e of an ellipse measures the ovalness of the ellipse. e = c/aIn the last example, what is the eccentricity?The smaller or closer to 0 that the eccentricity is, the morethe ellipse looks like a circle.The closer to 1 the eccentricity is, the more elongated it is.
13Find the center, vertices, and foci of the ellipse given by 4x2 + y2 - 8x + 4y - 8=0First, put this equation in standard form.4(x2 - 2x + 1) + ( y2 + 4y + 4) =4(x - 1)2 + (y + 2)2 = 16C( , )a =b =c =Vertices ( , ) ( , )Foci ( , ) ( , )e =Sketch it.
16The standard form with center (h,k) is Note: a is under the positive term. It is not necessarilytrue that a is bigger than b.
17Let’s take a look at the first hyperbola form. V(h-a,k)cV(h+a,k)bbC(h,k)F(h-c,k)F(h+c,k)ac is the distancefrom the centerto the foci.Note: If c isthe distancefrom the centerto F, and all radiiof a circle = ,then the hyp. of the right triangle is also c.Therefore, to find c,a2 + b2 = c2
18Sketch the hyperbola whose equation is 4x2 - y2 = 16. First divide by 16.Write down a, b, c and the center pt.a = 2b = 4Note: a is always underthe (+) term.C(0,0)Now find c.Let’s sketch the hyperbola.
19FFVVNow, we need to findthe equations of theasymptotes.What are their slopesand one point that is on both lines?
20Sketch the graph of4x2 - 3y2 + 8x +16 = 04(x2 + 2x ) - 3y2 = -16+1+ 44(x + 1)2 - 3y2 = -12Now, divide by -12 and switchthe x and y terms.C( , )a =b =c =e =Sketch