Presentation on theme: "The Beginning of Chapter 8:"— Presentation transcript:
1The Beginning of Chapter 8: Conic Sections (8.1a)Parabolas!!!
2V Imagine two non-perpendicular lines intersecting at point V. Rotating one of the lines (thegenerator) around the other (theaxis) yields a pair of rightcircular cones…A conic section is formed by theintersection of a plane and thesecones.VBasics: Parabola, Ellipse, HyperbolaDegenerates: Point, Line, IntersectingLinesCheck out the diagram on p.631…
3Conic Sections All conic sections can be defined algebraically as the graphs of second-degree (quadratic) equationsin two variables…in the form:where A, B, and C are not all zero.Today, we focus on parabolas!!!
4Definition: ParabolaA parabola is the set of all points in a plane equidistant from aparticular line (the directrix) and a particular point (the focus)in the plane.Point on the parabolaAxisDist. tofocusDist. todirectrixFocusVertexDirectrix
5Let’s equate these two distances: Deriving the equation of a parabolaFocus F(0, p)pP(x, y)pLet’s equate thesetwo distances:Directrix: y = –pD(x, –p)
6Deriving the equation of a parabola Standard form of the equation of an up- or down-openingparabola. If p > 0, it opens up, if p < 0, it opens down.
7Deriving the equation of a parabola Focus F(0, p)pP(x, y)pDirectrix: y = –pD(x, –p)The value p is the focal length of the parabola.A segment with endpoints on a parabola is a chord.The value |4p| is the focal width.
8Deriving the equation of a parabola Focus F(0, p)pP(x, y)pDirectrix: y = –pD(x, –p)Parabolas that open right or left are inverse relations of theupward or downward opening parabolas…standard form:
9Parabolas with Vertex (0, 0) Standard EquationOpensUpward ordownwardTo the right orto the leftFocusDirectrixAxisy-axisx-axisFocal LengthFocal Width
10Guided Practice Focus: (0, –1/2), Directrix: y = 1/2, Focal Width: 2 Find the focus, the directrix, and the focal width of the givenparabola. Then, graph the parabola by hand.Focus: (0, –1/2),Directrix: y = 1/2,Focal Width: 2
11Guided PracticeFind an equation in standard form for the parabola whosedirectrix is the line x = 2 and whose focus is the point (–2, 0).Would a graph help???y = –8x2
12Guided PracticeFind an equation in standard form for the parabola whosevertex is (0, 0), opens downward, and has a focal width of 4.Would a graph help???2x = – 4y
13Whiteboard Practice … Standard Form: Find an equation in standard form for the parabola with vertex(0, 0), opening upward, with focal width = 3.(since parabolaopens upward)Standard Form:
15We have only considered parabolas with the vertex on the origin………………… what happens when it’s not???VF (h, k + p)(h, k)F (h + p, k)V (h, k)Such translations do not change the focal length, the focalwidth, or the direction the parabola opens!!!
16Parabolas with Vertex (h, k) Standard EquationOpensUpward or downwardFocusDirectrixAxisFocal LengthFocal Width
17Parabolas with Vertex (h, k) Standard EquationOpensTo the right or to the leftFocusDirectrixAxisFocal LengthFocal Width
18Practice ProblemsFind the standard form of the equation for the parabola withvertex (3, 4) and focus (5, 4).Which general equation do we use?What are the values of h and k?Now, how do we find p?
19Practice Problems Use a function grapher to graph the given parabola. First, we must solve for y!!!Now, plug these two equationsinto your calculator!!!
20Practice ProblemsProve that the graph of the given equation is a parabola, thenfind its vertex, focus, and directrix.We need to complete the square…The CTS step!!!We have h = 2, k = –1,and p = 6/4 = 1.5Vertex: (2, –1), Focus: (3.5, –1), Directrix: x = 0.5
21Practice Problems Standard Form: Find an equation in standard form for the parabola that satisfiesthe given conditions.Vertex (–3, 3), opens downward, focal width = 20(since parabolaopens downward)Standard Form:
22Practice Problems Standard Form: Find an equation in standard form for the parabola that satisfiesthe given conditions.Vertex (2, 3), opens to the right, focal width = 5(since parabolaopens to the right)Standard Form: