Presentation on theme: "The Beginning of Chapter 8:"— Presentation transcript:
1 The Beginning of Chapter 8: Conic Sections (8.1a)Parabolas!!!
2 V 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…
3 Conic 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!!!
4 Definition: 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
5 Let’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)
6 Deriving 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.
7 Deriving 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.
8 Deriving 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:
9 Parabolas with Vertex (0, 0) Standard EquationOpensUpward ordownwardTo the right orto the leftFocusDirectrixAxisy-axisx-axisFocal LengthFocal Width
10 Guided 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
11 Guided 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
12 Guided 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
13 Whiteboard 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:
15 We 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!!!
16 Parabolas with Vertex (h, k) Standard EquationOpensUpward or downwardFocusDirectrixAxisFocal LengthFocal Width
17 Parabolas with Vertex (h, k) Standard EquationOpensTo the right or to the leftFocusDirectrixAxisFocal LengthFocal Width
18 Practice 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?
19 Practice Problems Use a function grapher to graph the given parabola. First, we must solve for y!!!Now, plug these two equationsinto your calculator!!!
20 Practice 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
21 Practice 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:
22 Practice 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: