Dte: 9. Conics: Prols Preclculus H. Notes: Unit 9 Conics Conic Sections: curves tht re formed y the intersection of plne nd doulenpped cone Syllus Ojectives:. The student will grph reltions or functions, including rel-world pplictions. (Prols) Prol: the conic formed y connecting ll points equidistnt from point (the focus) nd line (the directri). Verte: the point, tht lies on the is of symmetry hlfwy etween the focus nd directri Ais of Symmetry: the line of symmetry of prol tht psses through the verte. p : the distnce the focus nd directri re from the verte Focus: point on the is of symmetry of prol equidistnt from the verte s the directri. (lwys inside the prol) Directri: line perpendiculr to the is of symmetry equidistnt from the verte s the focus. Prol 4 py, p 0 Prol 4 py, p 0 p = distnce etween focus to directri nd verte Focl Width = 4p: the distnce cross the prol t the focus Prol y 4 p, p 0 Prol y 4 p, p 0 Pge of 5 Preclculus Topics in Anlytic Geometry: Lrson Chpter 9
Becuse p is negtive, the prol will open to the left. is of symmetry is horizontl: y 0 E : Sketch the grph of the prol Verte: 0,0 Becuse y is squred, Find p: y 3 Preclculus H. Notes: Unit 9 Conics 3 y. Focus: Directri: EX : Find n eqution of the prol with the verte t the origin.. Focus (3,0). Focus (0, -3) c. Directri y = - c. Directri = - EX 3: Find the stndrd form of the eqution of the prol. Find the focus. Pge of 5 Preclculus Topics in Anlytic Geometry: Lrson Chpter 9
Preclculus H. Notes: Unit 9 Conics Prol: center not (0,0) opens left or right: y k 4p h Verte: hk, Focus: h p, k Directri: h p opens up or down: h 4p y k Verte: hk, Focus: h, k p Directri: y k p E 4: Sketch the grph of the prol 7 8 y 3 Becuse it is in the form h 4p y k. Identify the verte, focus, nd directri. (,3), the prol opens: p is positive, so the prol opens up. Find p: Verte: Focus: p units up from the verte Directri: horizontl line p units down from the verte E 5: Write n eqution of the prol whose verte is t 3, nd whose focus is t 4,. Begin with sketch. The prol opens towrd the focus, so it opens right. Find p: The distnce from the focus to the verte is. The prol opens right, so p =. We will use the eqution y k 4p h. The verte (h, k) is, nd p = The eqution is: Pge 3 of 5 Preclculus Topics in Anlytic Geometry: Lrson Chpter 9
Preclculus H. Notes: Unit 9 Conics Proloid of Revolution: the result of rotting prol in three-dimensionl spce out its is Emples flshlights, hedlights, serchlights, stellite dishes, etc. Appliction Prolem Proloid E6: A stellite dish is 8 feet in dimeter. If it hs depth of 0.5 feet, how fr from the verte should the receiving ntenn e plced? (Hint: The ntenn should e plced t the focus.) A prol tht represents the sitution hs verte of 0,0 nd psses through the point. Grph: Eqution of the prol: 4,0.5 : The focus is ft from the verte. h 4p y k 4py EX 7: A store uses prolic mirror to see ll of the isles in the store. A cross section of the mirror is shown. Write n eqution for the cross section of the mirror nd identify the focus. We will use the eqution h 4p y k. The verte is 0,0 hk,, so we now hve 4 The prol psses through the point, y, 4 py 4 p p p The eqution is up from the verte: 4 y y py.. Use this to find p.. The focus is p units EX 8 Find the stndrd form of prol: squre. Then grph. y 4 3 0 y completing the You Try: Write n eqution in stndrd form of the prol with focus of, nd directri. Hint: Rememer tht the verte is hlfwy etween the focus nd the directri. Reflection: Eplin the reltionship etween the verte, focus, nd directri nd how these help you determine the orienttion of the prol Pge 4 of 5 Preclculus Topics in Anlytic Geometry: Lrson Chpter 9
Dte: 9. Conics: Ellipse Preclculus H. Notes: Unit 9 Conics Conic Sections: curves tht re formed y the intersection of plne nd doulenpped cone Syllus Ojectives:. The student will grph reltions or functions, including rel-world pplictions. (Ellipses) Ellipse: the set of ll points P such tht the sum of the distnces etween P nd two distinct fied points is constnt. Foci: the two fied points tht crete n ellipse Vertices: the two points t which the line through the foci intersect the ellipse Mjor Ais: the line segment joining the two vertices Center of the Ellipse: the midpoint of the mjor is Co-Vertices: the two points t which the line perpendiculr to the mjor is t the center intersects the ellipse Minor Ais: the line segment joining the co-vertices *** is lwys the iggest numer in ellipses! Horizontl Mjor Ais: UFO Verticl Mjor Ais: Footll y Mjor Ais Horizontl Vertices: (, 0) (-, 0) Foci: (c, 0) (-c, 0) Mjor Ais: Minor Ais: Distnce from foci to center: c y Mjor Ais Verticl Vertices: (0, ) (0, -) Foci: (0, c) (0, -c) Mjor Ais: Minor Ais: Pge 5 of 5 Preclculus Topics in Anlytic Geometry: Lrson Chpter 9
Preclculus H. Notes: Unit 9 Conics c Eccentricity: vlue tht descries the roundness of the ellipse e In n ellipse, 0e. If e 0, the grph is circle. If e, the grph is line Don t confuse this e with the numer e used with eponentil nd logrithmic equtions! Eqution of n Ellipse: h y k or h y k w/center t (h,k) E : Drw the ellipse given y 4 5y 00. Identify the foci. E. Drw the ellipse given y y. Identify the foci. 6 36 E 3: Write n eqution of the ellipse with center t co-verte t,0. 0,0, verte t 0, 3, nd Pge 6 of 5 Preclculus Topics in Anlytic Geometry: Lrson Chpter 9
Preclculus H. Notes: Unit 9 Conics E. 4: Write the eqution of the ellipse with Foci 5,0 nd mjor is of length. E5: Grph the ellipse nd stte its eccentricity. Find the foci. y Center hk, :, Mjor is is verticl ecuse is under the y-term. Vertices:, nd, Covertices:, nd, Length of Mjor Ais: Foci: c Eccentricity: nd, c, c c e Grph: Length of Minor Ais: 9 5 Trnsform Generl Form to Stndrd Form E6: Write the eqution in stndrd form. Complete the squre: 9 5y 36 0y 36 0 Set equl to : Writing n Eqution of n Ellipse E7: Write n eqution of n ellipse if the foci re 0, 3 nd length of 3. Center hlfwy etween the foci: 0,0 Mjor Ais: Distnce from center to focus: c 0,3 nd the mjor is hs c The foci re on the mjor is, so the mjor is is verticl ( under y-term). h y k : Pge 7 of 5 Preclculus Topics in Anlytic Geometry: Lrson Chpter 9
Preclculus H. Notes: Unit 9 Conics E8: Write n eqution of n ellipse if focus is 0, nd coverte is 3,3. Coverte endpoint of the minor is. Reflect over the mjor is to find the other coverte,. Focus reflect over the minor is to find the other focus,. Center: Intersection of the mjor nd minor es. Distnce etween center nd coverte: Distnce etween center nd focus: c c Eqution (mjor is is horizontl): h y k Appliction Prolem Orits: plnets, steroids, nd comets ll follow ellipticl pths s they orit the Sun. E9: The orit of Hlley s comet is 36.8 AU long nd 9. AU wide. Write n eqution tht represents its ellipticl orit. Note: An AU (stronomicl unit) is unit of length equl to ppro. 50 million km (93 million miles). It is the men distnce etween the Erth nd the Sun over one Erth orit. Mjor is: Minor Ais: y Eqution: Center: You Try: Grph the ellipse. Stte the eccentricity nd foci. y 6 4 Reflection: A whispering gllery is shped like n ellipse. If person stnds t one focus nd whispers, he cn e herd clerly y person stnding t the other focus. Why? Pge 8 of 5 Preclculus Topics in Anlytic Geometry: Lrson Chpter 9
9.3 Conics: Hyperol Preclculus H. Notes: Unit 9 Conics Hyperol: the set of ll points P such tht the difference of the distnces from P to two fied points, clled the foci is constnt Vertices: the two points t which the line through the foci intersects the hyperol Trnsverse Ais: the line segment joining the vertices Distnce etween foci: c Center: the midpoint of the trnsverse is Distnce from foci to center: Note: The foci of the hyperol lie on the trnsverse is, c units from the center where c. Hyperols hve slnt symptotes. Drw the rectngle. The lines tht pss through the corners of this rectngle re the slnt symptotes of the hyperol. Horizontl Trnsverse Ais: Hurley Verticl Trnsverse Ais: Hppy/Sd y y : Vertices: (, 0) (-, 0) Vertices: (0, ) (0, -) Foci: (c, 0) (-c, 0) Foci: (0, c) (0, -c) Asymptotes: Asymptotes: Distnce from foci to center: Pge 9 of 5 Preclculus Topics in Anlytic Geometry: Lrson Chpter 9
Preclculus H. Notes: Unit 9 Conics E. : Find foci, vertices, nd equtions of the symptotes for the hyperol. E. : Find foci, vertices, nd equtions of the symptotes for the hyperol. E. 3: Write the eqution of hyperol with foci t 0, vertices t 0, nd 0,. nd 0, nd Asymptotes: hyperol hs two slnt symptotes y k h (If hyperol opens left nd right, = rise, = run.) or y k h (If hyperol opens left nd right, = rise, = run.) Eccentricity: c e For hyperol, e. Pge 0 of 5 Preclculus Topics in Anlytic Geometry: Lrson Chpter 9
Preclculus H. Notes: Unit 9 Conics Grphing Hyperol with center t (h,k) E4: Sketch the grph of the hyperol. Stte the center, vertices, foci, symptotes, nd y eccentricity. 6 9 Center: hk, Hyperol opens nd. Vertices re to the right nd left of the center. : c Foci re c units to the right nd left of the center. c Asymptotes: y k h Eccentricity: e Grph: (Drwing rectngle with the symptotes s digonls helps to sketch the grph.) E5: Sketch the grph of the hyperol. Stte the center, vertices, foci, symptotes, nd eccentricity. Center: hk, y 6 4 Vertices re ove nd elow center. : c Hyperol opens. Foci re c units ove nd elow the center. Asymptotes: y k h c Eccentricity: e Grph: (Drwing rectngle with the symptotes s digonls helps to sketch the grph.) Pge of 5 Preclculus Topics in Anlytic Geometry: Lrson Chpter 9
Preclculus H. Notes: Unit 9 Conics Trnsform Generl Form to Stndrd Form E6: Write the stndrd eqution of Complete the squre: 4y 6y 0. Set equl to : more prctice: Center: Vertices: Foci: Asymptotes: Eccentricity: Grph: Writing n Eqution of Hyperol E7: Write n eqution of hyperol if the foci re re 4, nd 4,9. 4,0 nd 4,0 nd the vertices Center is hlfwy etween the foci (or the vertices): = distnce etween center nd vertices: c = distnce etween center nd foci: c c Opens up nd down y k h Pge of 5 Preclculus Topics in Anlytic Geometry: Lrson Chpter 9
Preclculus H. Notes: Unit 9 Conics E8: Write n eqution of hyperol if the vertices re through the point 5,4., nd, nd it psses Center is hlfwy etween the vertices: = distnce etween center nd vertices: Opens left nd right: h y k Sustitute y, nd solve for : Eqution: y 4 You Try: Write n eqution of hyperol if the foci re 0, 3 nd the eqution of n symptote is y 3. QOD: Eplin why the eccentricity of hyperol must e greter thn. Pge 3 of 5 Preclculus Topics in Anlytic Geometry: Lrson Chpter 9
Preclculus H. Notes: Unit 9 Conics Dte 9.8 PolrEqutions of Conics Syllus Ojective:.8 The student will construct the grph of function under given trnsltion, diltion, or reflection (or rottion). Review: Which conic is it? Complete the squre: 4 9y 6 8y 0 Set equl to : Grphing on the Clcultor E: Grph the ellipse from the emple ove on the clcultor. To grph on the clcultor, we must solve for y: for y. y y Alwys use stndrd form to solve 4 4 9 4 4 9 9 4 y 4 y 9 9 We must grph these s two seprte functions on the clcultor: y Note: It is clcultor error tht prevents it from enclosing the ellipse. Discriminnt Test: If n eqution is written in the form the disriminnt is B B B B 4AC. 4AC 0 Hyperol 4AC 0 Prol 4AC 0 Ellipse A By Cy D Ey F 0, then E: Verify tht the conic in the review is n ellipse using the discriminnt. 4 9y 6 8y 0 A 4, B 0, C 9 B 4 AC 0, so! Pge 4 of 5 Preclculus Topics in Anlytic Geometry: Lrson Chpter 9
Preclculus H. Notes: Unit 9 Conics 3y 4 0 A 0, B 3, C 0 E3: Identify the conic using the discriminnt. Then solve for y nd grph. B 4 AC 0, so 4 3y 4 0 y 3 E4: Identify the type of conic, write the eqution in stndrd form, y 8 6y 3 0 Discriminnt: A 0, B 0, C Stndrd form: B y 8 6y 3 0 4 AC 0, so You Try: Identify the type of conic, write the eqution in stndrd form y 8 6y 3 0 Reflection: How is the discriminnt of conic section in generl form different from the discriminnt used with qudrtic equtions (prols)? Pge 5 of 5 Preclculus Topics in Anlytic Geometry: Lrson Chpter 9