9.1 apply the distance and midpoint formulas

Transcription

1 9.1 pply the distnce nd midpoint formuls DISTANCE FORMULA MIDPOINT FORMULA To find the midpoint between two points x, y nd x y 1 1,, we Exmple 1: Find the distnce between the two points. Then, find the midpoint of the line segment joining the two points. Round ll non-integer nswers to two deciml plces.. 1.5, 4 nd.3, 9 Distnce: Midpoint:, Wht re the differences between: Sclene Tringle Isosceles Tringle Equilterl Tringle Exmple : The vertices of tringle re given. Clssify the tringle s sclene, isosceles, or equilterl. 4, 1, 6 0, 1 It my be helpful to lbel the points! Distnce: Type of Tringle: Distnce: Distnce:

2 Exmple 3: Use the given distnce d between the two points to find the vlue of x., 7 x, 1 d 89 Exmple 4: Write n eqution for the perpendiculr bisector of the line segment joining the two points: 3, 4 D5, 6 C Remember to mke n eqution for line, we need point nd slope! Point: Find the midpoint of the segment. Slope: Find the slope of the segment, then pply the opposite reciprocl. Plce the Point nd Slope into y y mx 1 x 1

3 9. Grph nd Write Equtions for Prbols ( x ) 1. Eqution : S. F.: x x 8y 4 py Step 1: Get the squred vrible or lone. x 8y Step : Identify the vertex. If there re no numbers being dded to or subtrcted from the vribles, you cn ssume the vertex is t 0,0. Vertex, Step 3: Identify the p vlue. To do so, set whtever is in front of the non-squred vrible equl to 4p. Solve for p. 8 4p Step 4: Determine the direction of opening. If p is positive, the grph will open up. If p is negtive, it will open down. Step 5: Find the focus. The focus is point found by dding p to the y vlue of the vertex. It is lwys locted within the rc of the prbol. (lwys dd) Step 6: Find the directrix. The directrix is horizontl line y. To find the number, subtrct the p vlue from the y vlue of the vertex. (lwys subtrct) Step 7: Determine the xis of symmetry. The xis of symmetry will be verticl line x. It will be equl to the x-coordinte of the vertex. Opens Focus y x, Step 8: Grph this informtion long with two dditionl points to complete the grph. x y

4 . Eqution : x 36y 0 x y 3. Eqution : S. F.: x 8y 3 x h 4 py k x y

5 9. Dy : Grph nd Write Equtions for Prbols ( y ) 1. Eqution : S. F.: y y 3x 4 px Step 1: Get the squred vrible or lone. y 3x Step : Identify the vertex. If there re no numbers being dded to or subtrcted from the vribles, you cn ssume the vertex is t 0,0. Vertex, Step 3: Identify the p vlue. To do so, set whtever is in front of the nonsqured vrible equl to 4p. Solve for p. 3 4p Step 4: Determine the direction of opening. If p is positive, the grph will open right. If p is negtive, the grph will open left. Opens Step 5: Find the focus. The focus is point found by dding p to the x vlue of the vertex. Focus, Step 6: Find the directrix. The directrix is verticl line x. To find the number, subtrct the p vlue from the x vlue of the vertex. Step 7: Determine the xis of symmetry. The xis of symmetry will be horizontl line y. It will be equl to the y-coordinte of the vertex. x y Step 8: Grph this informtion long with two dditionl points to complete the grph. x y

6 . Eqution : 3y 4x x y 3. Eqution : S. F.: y 3 0x 4 y k 4 px h x y

7 9. Dy 3 (continued) Exmple 1: Write the stndrd form of the eqution of the prbol with the given focus 0, 4 nd vertex t, 0 0. Grph the given informtion to help you determine whether this would be n x 4py or y 4px eqution. Determine the p vlue. Plce the p vlue into the stndrd form nd simplify the eqution. 5 Exmple : Write the stndrd form of the eqution of the prbol with the given focus, 0 nd vertex t 0, 0. Grph the given informtion to help you determine whether this would be n x 4py or y 4px eqution. Determine the p vlue. Plce the p vlue into the stndrd form nd simplify the eqution. Exmple 3: Write the stndrd form of the eqution of the prbol with the given directrix 5 x nd vertex t, 0 0. Grph the given informtion to help you determine whether this would be n x 4py or y 4px eqution. Determine the p vlue. Plce the p vlue into the stndrd form nd simplify the eqution.

8 9.3 Grph nd Write equtions of circles Notice tht x nd Exmple 1: Eqution : 5y 5x 40 y re on the left side of the eqution, joined by plus sign, nd tht the x is listed first. Also notice tht the x nd y re lone (no numbers in front). 5y 5x 40 Step 1: Get the squred vribles or with the x listed first. on the left side of the equl sign Step : Identify the center. If there re no numbers being dded to or subtrcted from the vribles, you cn ssume the center is t 0,0. Center, Step 3: Identify the rdius. To do so, tke the squre root of the number on the right side of the equl sign. (no decimls) r = Step 4: Grph the center long with severl points, s determined by the rdius. x y Wht is the difference between the stndrd eqution of circle nd the stndrd eqution of the prbols?

9 S. F.: Exmple : Eqution : x h y k r y 1 x 5 64 x y Exmple 3: Write the stndrd form of the eqution of the circle with the given rdius r 5 nd whose center is t 7,1. x h y k S. F.: r Eqution: Exmple 4: Write the stndrd form of the eqution of the circle tht psses through the given point 8,14 nd whose center is the origin. x h y k S. F.: r Eqution:

10 9.4 Grph nd Write Equtions OF ELLIPSES An ellipse is the set of ll points P in plne such tht the sum of the distnces between P nd two fixed points, clled the foci, is constnt. Stndrd form of n ellipse with horizontl mjor xis. x h y k b 1 In ech problem, we will grph the center, the vertices, the co-vertices, nd the foci.

11 Eqution : 4x 5y 100 4x 5y x h y k S. F.: Step 1: Get the right side of the eqution equl to 1. b 1 Step : Identify the center. If there re no numbers being dded to or subtrcted from the vribles, you cn ssume the center is t0,0. If there re numbers being dded or subtrcted, tke the opposite of ech s the x nd y coordintes of the center. Center, Step 3: Determine which denomintor contins the. The is the lrger denomintor. The purpose of the vlue is to help us crete the mjor xis nd the vertices tht re on ech end of the mjor xis. Tody, the is under the x vrible, which mens tht the mjor xis will be horizontl like the x-xis. Find the nd determine how the mjor xis will look. Tke the squre root of to determine. Mjor xis: = = Step 4: The vertices re points t either end of the mjor xis. If the is under the x- vrible, the vertices cn be found by dding to the x vlue of the center. Vertices,, Step 5: Determine which denomintor contins the b. It will lwys be the smller of the two denomintors. It will help us determine the minor xis nd the co-vertices tht re on ech end. Tody, the b is under the y vrible, which mens tht the minor xis will be verticl like the y -xis. Find the b nd use it to determine b. Step 6: The co-vertices re points t either end of the minor xis. Since the b is under the y -vrible, the vertices cn be found by dding b to the y vlue of the center. Covertices Step 7: Determine the foci. Foci re two points locted on the mjor xis. c 5 4 Becuse, tody, the mjor xis is horizontl like the x-xis, we will find these c 1 two points by dding c to the x vlue of the center. C is determined by the formul: c b c 1 or Foci c 5,,,, Step 8: Grph the center, vertices, line long the mjor xis, co-vertices, line long the minor xis, nd foci. Use the mjor nd minor xis lines to help you sketch the ellipse.

12 Eqution :. x h y k S. F.: 9x 16y 144 b 1 Center, Mjor xis: = Vertices,, Co-vertices,, c = Foci,,

13 3. x 5 4y x 1 y 3 8 Center, Mjor xis: Center, Mjor xis: = Vertices,, Vertices,, Co-vertices,, Co-vertices,, c = Foci,, Foci,,

14 9.4 Grph nd Write Equtions OF ELLIPSES (dy ) An ellipse is the set of ll points P in plne such tht the sum of the distnces between P nd two fixed points, clled the foci, is constnt. Stndrd form of n ellipse with verticl mjor xis. y k x h b 1 In ech problem, we will grph the center, the vertices, the co-vertices, nd the foci.

15 Eqution : 5x 4y 100 5x 4y y k x h S. F.: Step 1: Get the right side of the eqution equl to 1. b 1 Step : Identify the center. If there re no numbers being dded to or subtrcted from the vribles, you cn ssume the center is t 0,0. If there re numbers being dded or subtrcted, tke the opposite of ech s the x nd y coordintes of the center. Center, Step 3: Determine which denomintor contins the. The is the lrger denomintor. The purpose of the vlue is to help us crete the mjor xis nd the vertices tht re on ech end of the mjor xis. Tody, the is under the y vrible, which mens tht the mjor xis will be verticl like the y -xis. Find the nd determine the how the mjor xis will look. Tke the squre root of to determine. Mjor xis: = = Step 4: The vertices re points t either end of the mjor xis. If the is under the y - vrible, the vertices cn be found by dding to the y vlue of the center. Vertices,, Step 5: Determine which denomintor contins the b. It will lwys be the smller of the two denomintors. It will help us determine the minor xis nd the co-vertices tht re on ech end. Tody, the b is under the x vrible, which mens tht the minor xis will be horizontl like the x -xis. Find the b nd use it to determine b. Step 6: The co-vertices re points t either end of the minor xis. Since the b is under the x -vrible, the vertices cn be found by dding b to the x vlue of the center. Covertices Step 7: Determine the foci. Foci re two points locted on the mjor xis. c 5 4 Becuse, tody, the mjor xis is verticl like the y-xis, we will find these two c 1 points by dding c to the y vlue of the center. C is determined by the formul: c b c 1 or Foci c 5,,,, Step 8: Grph the center, vertices, line long the mjor xis, co-vertices, line long the minor xis, nd foci. Use the mjor nd minor xis lines to help you sketch the ellipse.

16 x y x 4 y Center, Mjor xis: Center, Mjor xis: = Vertices,, Vertices,, Co-vertices,, Co-vertices,, c = Foci,, Foci,,

17 9.5 Grph nd Write Equtions OF hyperbols A hyperbol is the set of ll point P such tht the difference of the distnces between P nd two fixed points, gin clled the foci, is constnt. Stndrd form of hyperbol with horizontl trnsverse xis. x h y k b 1 In ech problem, we will grph the center, the vertices, line through the trnsverse xis, the conjugte xis, the foci, nd the symptotes.

18 Eqution : 5x 4y 100 5x 4y x h y k S. F.: b 1 Step 1: Get the right side of the eqution equl to 1. Step : Identify the center nd grph it. Center, Step 3: Determine which denomintor contins the. The is lwys underneth the positively squred vrible. The purpose of the vlue is to help us crete the trnsverse xis nd the vertices tht re on ech end of the trnsverse xis. Tody, the is under the x vrible, which mens tht the trnsverse xis will be horizontl like the x-xis. Find the nd tke the squre root of to determine. = = Step 4: The vertices re points t either end of the trnsverse xis. If the is under the x-vrible, the vertices cn be found by dding to the x vlue of the center. Drw the vertices nd connect them to crete the trnsverse xis. Vertices,, Step 5: Determine which denomintor contins theb. It will lwys be below the negtively squre vrible. Tody, the b is under the y vrible. Find the b nd use it to determine b. Conjugte xis points: Step 6: The b vlue will help us crete the two points tht connect the conjugte xis. Since the b is under the y -vrible, the points cn be found by dding b to the y vlue of the center. Step 7: Determine the foci. Foci re two points locted beyond the trnsverse xis. Becuse, tody, the trnsverse xis is horizontl like the x-xis, we will find these two points by dding c to the x vlue of the center. C is determined by the formul: c b Step 8: Determine the symptotes. Wht s unique bout hyperbols is the presence of symptotes. For tody s hyperbols, in which the is under the x- b vrible, the eqution will be. We will strt from the center of the hyperbol nd b b pply this eqution s two different slopes: nd. We will drw the symptotes s dshed lines. They will help us crete the shpe of the brnches. Foci b 5 5,,,, nd 5 Step 9: Grph the center, the vertices, line through the trnsverse xis, the conjugte xis, the foci, nd the symptotes. Then sketch the shpe of the brnches.

19 . 4x 49y 196 = c = Asymptotes: Center Vertices Conjugte xis points Foci,,,,,,, 3. 4x y 36 Center, = Vertices,, b = Conjugte xis points,, c = Foci,, Asymptotes:

20 9.5 Grph nd Write Equtions OF hyperbols (Dy ) Stndrd form of hyperbol with verticl trnsverse xis. y k x h b 1 In ech problem, we will grph the center, the vertices, line through the trnsverse xis, the conjugte xis, the foci, nd the symptotes. 1. Eqution : 4y 16x 64 4y 16x 64 Step 1: Get the right side of the eqution equl to 1. Step : Identify the center nd grph it. Center, Step 3: Determine which denomintor contins the. Tody, the is under the y vrible, which mens tht the trnsverse xis will be verticl like the y-xis. = = Step 4: The vertices re points t either end of the trnsverse xis. If the is under the y-vrible, the vertices cn be found by dding to the y vlue of the center. Vertices Step 5: Determine which denomintor contins theb. It will lwys be below the negtively squre vrible. Tody, the b is under the x vrible. Step 6: The b vlue will help us crete the two points tht connect the conjugte xis. Since the b is under the x -vrible, the points cn be found by dding b to the x vlue of the center. Step 7: Determine the foci. Becuse, tody, the trnsverse xis is verticl like the y-xis, we will find these two points by dding c to the y vlue of the center. C is determined by the formul: c b Conjugte xis points: Foci Step 8: Determine the symptotes. For tody s hyperbols, in which the is under the y-vrible, the eqution will be. b b,,,,,, Step 9: Grph the center, the vertices, line through the trnsverse xis, the conjugte xis, the foci, nd the symptotes. Then sketch the shpe of the brnches.

21 . 9y 36x 36 Center, = Vertices,, Conjugte xis points,, c = Foci,, Asymptotes: 3. 10y 5x 100 Center, Vertices,, Conjugte xis points,, Foci,, Asymptotes:

22 9.6 clssify conic sections Any conic cn be described by generl second degree eqution, where A, B, C, D, E, nd F re just coefficients. Given ny conic eqution, we cn figure out whether it is Prbol, Circle, n Ellipse, or Hyperbol. Ax Bxy Cy Dx Ey F 0 1. x y x 4y 1 0 CTS: 1. Isolte the x s nd y s with prenthesis. If you hve both the squred nd liner version of the vrible, then you ll need to complete the squre.. Mke sure tht the coefficient on the squred vrible is 1. If not, pull out GCF. b 3. Find the c vlue with the eqution 4. Crefully blnce the eqution. 5. Fctor. 6. Bsed on wht you know bout the conic, rewrite it in its stndrd form..

23 . x y 4x x y 16x 4y y 4y x 6 0