Learning About Conic Sections
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Learning about Conic Sections
As
pt ot e
Hyper bolas
Vertex •• Foci
As
ym
Axis
e
Transverse
t pto ym
Conjugate
•Foci • Vertex
Remember: Transverse axis always cross the Foci. Conjugate axis do not
Axis
Center- (h, k) The equation of hyperbolas are
and
Horizontal
Vertical
To find Foci c is found by a²+b²=c² (Foci is always constant) Asymptote equation are Horizontal hyperbola
and Vertical Hyperbola
Asymptote equations correspond with each kind of hyperbola
Ex. 1 Equation x²/16-y²/4=1 •
•
Center- (0,0) x² is put first so the hyperbola is horizontal.
Step One- Find a and b, a is 4 units left and right, b is 2 units up and down. √16= 4 √4=2 Step Two- Using the asymptote equation you’re given y=±½x -Graph it Step Three-Use foci equation c²=16+4→c²=20→c 4.5
Foci-(±4.5,0)
Step Four- Vertexes are at (±4,0) since a is 4 and x value of the center is 0 When graphing the hyperbola curves the parabolas never touch the asypt.
Ex. 2 y - 2 = (-6/5)(x - 1)
Equation-
y - 2 = (+6/5)(x - 1) F V
Center- (1,2) C
(y-2)² is set as first term so it is a vertical hyperbola.
V F
Step One- Given the center (1,2) find a and b and graph the box a.√36=6
b.√25=5
Step Two-Find Asymptotes’ equation, y=6/5x+4/5 and y=6/5x+16/5 Step Three- Use c²=b²+a² to find foci
c²=36+25
c=√61
c 7.8
(1,9.8) (1,-5.8)
Step Four-to find Vertices add & subtract 6 from the center y value (1,8) (1,4) Remember- When the center is not (0,0) asymptote equation has y-intercepts written
Work Prob. GRAPH and give the center, vertices, equation of asymptotes, and the lengths of a, b.
Work Prob. GRAPH and give the center, vertices, equation of asymptotes, and the lengths of a, b. Step One. Find the center and a, b to begin graphing the box. There are no numbers after x² and y² so center is (0,0)
Work Prob. GRAPH and give the center, vertices, equation of asymptotes, and the lengths of a, b. Step One. Find the center and a, b to begin graphing the box. There are no numbers after x² and y² so center is (0,0)
a²=25 b²=9 a=5 NOW GRAPH IT!
b=3
Work Prob. GRAPH and give the center, vertices, equation of asymptotes, and the lengths of a, b.
C
Work Prob. GRAPH and give the center, vertices, equation of asymptotes, and the lengths of a, b. Step Two- Find the asymptotes
C
Y=±b/ax because it is a horizontal hyperbola it is b/a Y=±3/5x
Graph it
Work Prob. GRAPH and give the center, vertices, equation of asymptotes, and the lengths of a, b. Step Two- Find the asymptotes
C
Y=±b/ax because it is a horizontal hyperbola it is b/a Y=±3/5x
Graph it
That is the asymptotic equation
Work Prob. GRAPH and give the center, vertices, equation of asymptotes, and the lengths of a, b. Step Two- Find the asymptotes Y=±b/ax because it is a horizontal hyperbola it is b/a
C
Y=±3/5x
Graph it
That is the asymptotic equation Step Three- Finding the foci to find it use c²=a²+b² Using a=5 & b=3
c²=5²+3²
c²=25+9
Add & subtract 5.8 from the center x-value giving Graph the Foci Points
√34=c (±5.8,0)
C=5.8
Work Prob. GRAPH and give the center, vertices, equation of asymptotes, and the lengths of a, b. Step Four- Graphing the hyperbola and finding vertices F •
C
F •
To find the vertices add & subtract a(5) to the x value of (0,0) by doing this you find the end of the hyperbola box. Now just graph the parabolic shape on each end of the box and make sure the parabola never touches the asymptotes.
Work Prob. GRAPH and give the center, vertices, equation of asymptotes, and the lengths of a, b. Step Four- Graphing the hyperbola and finding vertices F •
C
F •
To find the vertices add & subtract a(5) to the x value of (0,0) by doing this you find the end of the hyperbola box. Now just graph the parabolic shape on each end of the box and make sure the parabola never touches the asymptotes. That’s All
Work Prob. Graph
Work Prob. Graph Step one- The equation is not in standard form of a hyperbola so to do that group common variables together and move numerical terms to other side.
Work Prob. Graph Step one- The equation is not in standard form of a hyperbola so to do that group common variables together and move numerical terms to other side. You should get
Work Prob. Graph Step one- The equation is not in standard form of a hyperbola so to do that group common variables together and move numerical terms to other side. You should get In order to have perfect squares you must use completing the square for (y²-6y) and (x²+2x)
Work Prob. Graph Step one- The equation is not in standard form of a hyperbola so to do that group common variables together and move numerical terms to other side. You should get In order to have perfect squares you must use completing the square for (y²-6y) and (x²+2x) By doing that, [y²-6y+(3²)] giving (y-3)²
Work Prob. Graph Step one- The equation is not in standard form of a hyperbola so to do that group common variables together and move numerical terms to other side. You should get In order to have perfect squares you must use completing the square for (y²-6y) and (x²+2x) By doing that, [y²-6y+(3²)] giving (y-3)² And [x²+2x+(1²)] giving (x+1)²
Work Prob. Graph Step one- The equation is not in standard form of a hyperbola so to do that group common variables together and move numerical terms to other side. You should get In order to have perfect squares you must use completing the square for (y²-6y) and (x²+2x) By doing that, [y²-6y+(3²)] giving (y-3)² And [x²+2x+(1²)] giving (x+1)²
Now just divide (11+9-4) on both side to get
Work Prob. Now Graph The center is (3,-1) because (y-3) and (x+1) Now find a, b to make the box
Work Prob. Now Graph The center is (3,-1) because (y-3) and (x+1) Now find a, b to make the box a.√16=4
b.√4=2
Work Prob. Now Graph The center is (3,-1) because (y-3) and (x+1) Now find a, b to make the box a.√16=4
b.√4=2
Step Two- Find the asymptote equations y-3=2(x-(±1)
y=2x+1 & y=2x+5
Work Prob. Now Graph The center is (3,-1) because (y-3) and (x+1) Now find a, b to make the box a.√16=4
b.√4=2
Step Two- Find the asymptote equations
Work Prob. Now Graph The center is (3,-1) because (y-3) and (x+1) Now find a, b to make the box a.√16=4
b.√4=2
Step Two- Find the asymptote equations y-3=2(x-(±1)
y=2x+1 & y=2x+5
Step Three- Find the foci
Work Prob. Now Graph The center is (3,-1) because (y-3) and (x+1) Now find a, b to make the box a.√16=4
b.√4=2
Step Two- Find the asymptote equations y-3=2(x-(±1)
y=2x+1 & y=2x+5
Step Three- Find the foci c²=4²+2²
c²=16+4
c=√20
c=4.5
Work Prob. Now Graph The center is (3,-1) because (y-3) and (x+1) Now find a, b to make the box a.√16=4
b.√4=2
Step Two- Find the asymptote equations y-3=2(x-(±1)
y=2x+1 & y=2x+5
Step Three- Find the foci c²=4²+2²
c²=16+4
c=√20
c=4.5
(1, 7.5) (1, -1.5) Now graph the center, hyperbola box, and the foci.
Work Prob. Now Graph Find the vertices of the hyperbola Add a(4) to the y value (-1, 3) •
•
•
Work Prob. Now Graph Find the vertices of the hyperbola Add a(4) to the y value (-1, 3) • • V
•
• V •
Work Prob. Now Graph Find the vertices of the hyperbola Add a(4) to the y value (-1, 3) • •
•
•
•
Now that the vertices are present draw the parabolas.
As
pt ot e
Hyper bolas
Vertex •• Foci
As
ym
Axis
e
Transverse
t pto ym
Conjugate
•Foci • Vertex
Remember: Transverse axis always cross the Foci. Conjugate axis do not
Axis
Center- (h, k) The equation of hyperbolas are
and
Horizontal
Vertical
To find Foci c is found by a²+b²=c² (Foci is always constant) Asymptote equation are Horizontal hyperbola
and Vertical Hyperbola
Asymptote equations correspond with each kind of hyperbola
Ex. 1 Equation x²/16-y²/4=1 •
•
Center- (0,0) x² is put first so the hyperbola is horizontal.
Step One- Find a and b, a is 4 units left and right, b is 2 units up and down. √16= 4 √4=2 Step Two- Using the asymptote equation you’re given y=±½x -Graph it Step Three-Use foci equation c²=16+4→c²=20→c 4.5
Foci-(±4.5,0)
Step Four- Vertexes are at (±4,0) since a is 4 and x value of the center is 0 When graphing the hyperbola curves the parabolas never touch the asypt.
Ex. 2 y - 2 = (-6/5)(x - 1)
Equation-
y - 2 = (+6/5)(x - 1) F V
Center- (1,2) C
(y-2)² is set as first term so it is a vertical hyperbola.
V F
Step One- Given the center (1,2) find a and b and graph the box a.√36=6
b.√25=5
Step Two-Find Asymptotes’ equation, y=6/5x+4/5 and y=6/5x+16/5 Step Three- Use c²=b²+a² to find foci
c²=36+25
c=√61
c 7.8
(1,9.8) (1,-5.8)
Step Four-to find Vertices add & subtract 6 from the center y value (1,8) (1,4) Remember- When the center is not (0,0) asymptote equation has y-intercepts written
Work Prob. GRAPH and give the center, vertices, equation of asymptotes, and the lengths of a, b.
Work Prob. GRAPH and give the center, vertices, equation of asymptotes, and the lengths of a, b. Step One. Find the center and a, b to begin graphing the box. There are no numbers after x² and y² so center is (0,0)
Work Prob. GRAPH and give the center, vertices, equation of asymptotes, and the lengths of a, b. Step One. Find the center and a, b to begin graphing the box. There are no numbers after x² and y² so center is (0,0)
a²=25 b²=9 a=5 NOW GRAPH IT!
b=3
Work Prob. GRAPH and give the center, vertices, equation of asymptotes, and the lengths of a, b.
C
Work Prob. GRAPH and give the center, vertices, equation of asymptotes, and the lengths of a, b. Step Two- Find the asymptotes
C
Y=±b/ax because it is a horizontal hyperbola it is b/a Y=±3/5x
Graph it
Work Prob. GRAPH and give the center, vertices, equation of asymptotes, and the lengths of a, b. Step Two- Find the asymptotes
C
Y=±b/ax because it is a horizontal hyperbola it is b/a Y=±3/5x
Graph it
That is the asymptotic equation
Work Prob. GRAPH and give the center, vertices, equation of asymptotes, and the lengths of a, b. Step Two- Find the asymptotes Y=±b/ax because it is a horizontal hyperbola it is b/a
C
Y=±3/5x
Graph it
That is the asymptotic equation Step Three- Finding the foci to find it use c²=a²+b² Using a=5 & b=3
c²=5²+3²
c²=25+9
Add & subtract 5.8 from the center x-value giving Graph the Foci Points
√34=c (±5.8,0)
C=5.8
Work Prob. GRAPH and give the center, vertices, equation of asymptotes, and the lengths of a, b. Step Four- Graphing the hyperbola and finding vertices F •
C
F •
To find the vertices add & subtract a(5) to the x value of (0,0) by doing this you find the end of the hyperbola box. Now just graph the parabolic shape on each end of the box and make sure the parabola never touches the asymptotes.
Work Prob. GRAPH and give the center, vertices, equation of asymptotes, and the lengths of a, b. Step Four- Graphing the hyperbola and finding vertices F •
C
F •
To find the vertices add & subtract a(5) to the x value of (0,0) by doing this you find the end of the hyperbola box. Now just graph the parabolic shape on each end of the box and make sure the parabola never touches the asymptotes. That’s All
Work Prob. Graph
Work Prob. Graph Step one- The equation is not in standard form of a hyperbola so to do that group common variables together and move numerical terms to other side.
Work Prob. Graph Step one- The equation is not in standard form of a hyperbola so to do that group common variables together and move numerical terms to other side. You should get
Work Prob. Graph Step one- The equation is not in standard form of a hyperbola so to do that group common variables together and move numerical terms to other side. You should get In order to have perfect squares you must use completing the square for (y²-6y) and (x²+2x)
Work Prob. Graph Step one- The equation is not in standard form of a hyperbola so to do that group common variables together and move numerical terms to other side. You should get In order to have perfect squares you must use completing the square for (y²-6y) and (x²+2x) By doing that, [y²-6y+(3²)] giving (y-3)²
Work Prob. Graph Step one- The equation is not in standard form of a hyperbola so to do that group common variables together and move numerical terms to other side. You should get In order to have perfect squares you must use completing the square for (y²-6y) and (x²+2x) By doing that, [y²-6y+(3²)] giving (y-3)² And [x²+2x+(1²)] giving (x+1)²
Work Prob. Graph Step one- The equation is not in standard form of a hyperbola so to do that group common variables together and move numerical terms to other side. You should get In order to have perfect squares you must use completing the square for (y²-6y) and (x²+2x) By doing that, [y²-6y+(3²)] giving (y-3)² And [x²+2x+(1²)] giving (x+1)²
Now just divide (11+9-4) on both side to get
Work Prob. Now Graph The center is (3,-1) because (y-3) and (x+1) Now find a, b to make the box
Work Prob. Now Graph The center is (3,-1) because (y-3) and (x+1) Now find a, b to make the box a.√16=4
b.√4=2
Work Prob. Now Graph The center is (3,-1) because (y-3) and (x+1) Now find a, b to make the box a.√16=4
b.√4=2
Step Two- Find the asymptote equations y-3=2(x-(±1)
y=2x+1 & y=2x+5
Work Prob. Now Graph The center is (3,-1) because (y-3) and (x+1) Now find a, b to make the box a.√16=4
b.√4=2
Step Two- Find the asymptote equations
Work Prob. Now Graph The center is (3,-1) because (y-3) and (x+1) Now find a, b to make the box a.√16=4
b.√4=2
Step Two- Find the asymptote equations y-3=2(x-(±1)
y=2x+1 & y=2x+5
Step Three- Find the foci
Work Prob. Now Graph The center is (3,-1) because (y-3) and (x+1) Now find a, b to make the box a.√16=4
b.√4=2
Step Two- Find the asymptote equations y-3=2(x-(±1)
y=2x+1 & y=2x+5
Step Three- Find the foci c²=4²+2²
c²=16+4
c=√20
c=4.5
Work Prob. Now Graph The center is (3,-1) because (y-3) and (x+1) Now find a, b to make the box a.√16=4
b.√4=2
Step Two- Find the asymptote equations y-3=2(x-(±1)
y=2x+1 & y=2x+5
Step Three- Find the foci c²=4²+2²
c²=16+4
c=√20
c=4.5
(1, 7.5) (1, -1.5) Now graph the center, hyperbola box, and the foci.
Work Prob. Now Graph Find the vertices of the hyperbola Add a(4) to the y value (-1, 3) •
•
•
Work Prob. Now Graph Find the vertices of the hyperbola Add a(4) to the y value (-1, 3) • • V
•
• V •
Work Prob. Now Graph Find the vertices of the hyperbola Add a(4) to the y value (-1, 3) • •
•
•
•
Now that the vertices are present draw the parabolas.
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