Section 9.3 The Parabola Objectives

1 900 Chapter 9 Conic Sections and Analytic as a quadraticequation in and then use the quadratic formula to express in terms of Graph the resulting two equations using agraphing utility in a by viewingrectangle. What effect does the have on the graph ofthe resulting hyperbola? What problems would youencounter if you attempted to write the given equation instandard form by completing the square? and in the same viewingrectangle. Explain why the graphs are not the Thinking ExercisesMake Sense?In Exercises 81 84, determine whether eachstatement makes sense or does not make sense, and explain changed the addition in an ellipse s equation to subtractionand this changed its elongation from horizontal to noticed that the definition of a hyperbola closely resemblesthat of an ellipse in that it depends on the distances betweena set of points in a plane to two fixed points, the graphed a hyperbola centered at the origin that hadbut no graphed a hyperbola centered at the origin that wassymmetric with respect to the and also symmetricwith respect to the ,x x 16-y y 9=1x216-y29=1xy-term3-30, 50, 1043-50, 70, +2y2-3x+10y-6=0In Exercises 85 88.

2 Determine whether each statement is true or the statement is false, make the necessary change(s) to produce atrue one branch of a hyperbola is removed from a graph, thenthe branch that remains must define as a function of points on the asymptotes of a hyperbola also satisfy thehyperbola s graph of does not intersect the line different hyperbolas can never share the same happens to the shape of the graph of the standard form of the equation of the hyperbola withvertices and (5, 6), passing through (0, 9). the equation of a hyperbola whose asymptotes ExercisesExercises 92 94 will help you prepare for the material covered inthe next Exercises 92 93, graph each Parabola with the given the terms involving on the left side of the equation:Then write the equation in an equivalent form by completingthe square on the left +2y+12x-23= +2y=x2+4x-515, -62ca:q, where c2=a2+b2?

3 X2a2-y2b2=1y=- 23 ParabolaThis NASA photograph is one of a series ofstunning images captured from the ends ofthe universe by the hubble SpaceTelescope. The image shows infantstar systems the size of our solarsystem emerging from the gas anddust that shrouded their a parabolic mirror that is in diameter, the hubble hasprovided answers to many of theprofound mysteries of the cosmos:How big and how old is theuniverse? How did the galaxiescome to exist? Do other Earth-like planets orbit other sun-like stars? In this Section , westudy parabolas and their applications, including parabolic shapes that gather distantrays of light and focus them into spectacular of a ParabolaIn Chapter 2, we studied parabolas, viewing them as graphs of quadratic functions inthe formy=a1x-h22+k or y=ax2+bx+ Graph parabolas with verticesat the origin.

4 Write equations of parabolasin standard form. Graph parabolas with verticesnot at the origin. Solve applied problemsinvolving first glance, this image looks like columns of smoke rising from afire into a starry sky. Those are, indeed, stars in the background, butyou are not looking at ordinary smoke columns. These stand almost6 trillion miles high and are 7000 light-years from Earth morethan 400 million times as far away as the 21-11-2008 13:28 Page 900 Section Parabola901 Study TipHere is a summary of what you should already know about graphing the graph opens upward. If the graph opens vertex of is of the vertex of is x=- +bx+cx-coordinateyy = a(x h)2 + k a > 0xxy(h, k)(h, k)x = hx = hy = a(x h)2 + k a < 01h, +ka60,a70,y ax2 bx cy a(x h)2 kParabolas can be given a geometric definition that enables us to includegraphs that open to the left or to the right, as well as those that open obliquely.

5 Thedefinitions of ellipses and hyperbolas involved two fixed points, the foci. Bycontrast, the definition of a Parabola is based on one point and a of a ParabolaA parabolais the set of all points in a plane that are equidistant from a fixedline, the directrix, and a fixed point, the focus, that is not on the line(seeFigure ).VertexDirectrixParabolaAxis ofsymmetryFocusFigure Figure , find the line passing through the focus and perpendicular to thedirectrix. This is the axis of symmetryof the Parabola . The point of intersection ofthe Parabola with its axis of symmetry is called the vertex. Notice that the vertex ismidway between the focus and the Form of the Equation of a ParabolaThe rectangular coordinate system enables usto translate a Parabola s geometric definitioninto an algebraic ourstarting point for obtaining an equation.

6 Weplace the focus on the at the pointThe directrix has an equation given byThe vertex, located midway betweenthe focus and the directrix, is at the does the definition of a parabolatell us about the point in Figure any point on the Parabola , thedistance to the directrix is equal to thedistance to the focus. Thus, the point is on the Parabola if and only ifUse the distance both sides of the equation. 1x+p22=1x-p22+y2 41x+p22+1y-y22=41x-p22+1y-022 d1= , y2d2d11x, y21x, y2x= , : x = pFocus (p, 0)xyM( p, y)P(x, y)d1d2 Figure 21-11-2008 13:28 Page 901902 Chapter 9 Conic Sections and Analytic GeometrySquare and Subtract from both sides of the for This last equation is called the standard form of the equation of a Parabola with itsvertex at the origin.

7 There are two such equations, one for a focus on the andone for a focus on the +p22px=-2px+ +px2+2px+p2=x2-2px+p2+y2 Standard Forms of the Equations of a ParabolaThe standard form of the equation of a parabolawith vertex at the origin isFigure (a)illustrates that for the equation on the left, the focus is on thewhich is the axis of (b)illustrates that for theequation on the right, the focus is on the which is the axis of ,x-axis,y2=4px or x2= TipIt is helpful to think of as thedirected distancefrom the vertex tothe focus. If the focus lies units to the right of the vertex or units above the vertex. If thefocus lies units to the left of thevertex or units below the vertex.

8 P p p60,ppp70,pUsing the Standard Form of the Equation of a ParabolaWe can use the standard form of the equation of a Parabola to find its focus anddirectrix. Observing the graph s symmetry from its equation is helpful in locatingthe the definition of a Parabola is given in terms of its focus and itsdirectrix, the focus and directrix are not part of the graph. The vertex, located at theorigin, is a point on the graph of and Example 1 illustrates howyou can find two additional points on the the Focus and Directrix of a ParabolaFind the focus and directrix of the Parabola given by Then graph 1x2= equation does not change ify is replaced with y. There isx-axis symmetry and the focus ison the x-axis at (p, 0).

9 The equation does not change ifx is replaced with x. There isy-axis symmetry and the focus ison the y-axis at (0, p). Graph parabolas with vertices atthe : x = pFocus (p, 0)Vertexxyy2 = 4pxFigure (a) Parabola with the as theaxis of symmetry. If the graph opens to theright. If the graph opens to the ,p70,x-axisDirectrix: y = pxyx2 = 4pyFocus (0, p)VertexFigure (b) Parabola with the as the axis of symmetry. If the graphopens upward. If the graph ,p70,y-axisP-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 902 Section Parabola903 SolutionThe given equation,is in the standard form soWe can find both the focus and the directrix by finding Divide both sides by is positive, the Parabola , with its symmetry, opens to the right.

10 Thefocus is 3 units to the right of the vertex, (0, 0).The focus, (3, 0), and directrix,are shown in Figure graph the Parabola , we will use two points on the graph that lie directlyabove and below the focus. Because the focus is at (3, 0), substitute 3 for in theparabola s equation,Replace with 3 in the square root points on the Parabola above and below the focus are (3, 6) and Thegraph is sketched in Figure Point1 Find the focus and directrix of the Parabola given by Then graph the general, the points on a Parabola that lie above and below the focus,are each at a distance from the focus. This is because if thenso The line segment joining these two points is called thelatus rectum; its length is 4p.