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HyperbolasJust as all the other conic sections, hyperbolas are definedby distance relations.
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HyperbolasJust as all the other conic sections, hyperbolas are definedby distance relations.Given two fixed points, called foci, a hyperbola is the setof points whose difference of the distances to the foci isa constant.
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HyperbolasJust as all the other conic sections, hyperbolas are definedby distance relations.Given two fixed points, called foci, a hyperbola is the setof points whose difference of the distances to the foci isa constant.If A, B and C are points on a hyperbola as shown C A B
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HyperbolasJust as all the other conic sections, hyperbolas are definedby distance relations.Given two fixed points, called foci, a hyperbola is the setof points whose difference of the distances to the foci isa constant.If A, B and C are points on a hyperbola as shown thena1 – a2 C A a1 a2 B
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HyperbolasJust as all the other conic sections, hyperbolas are definedby distance relations.Given two fixed points, called foci, a hyperbola is the setof points whose difference of the distances to the foci isa constant.If A, B and C are points on a hyperbola as shown thena1 – a2 = b1 – b2 C A a1 a2 b2 B b1
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HyperbolasJust as all the other conic sections, hyperbolas are definedby distance relations.Given two fixed points, called foci, a hyperbola is the setof points whose difference of the distances to the foci isa constant.If A, B and C are points on a hyperbola as shown thena1 – a2 = b1 – b2 = c2 – c1 = constant. C c2 A a1 c1 a2 b2 B b1
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HyperbolasA hyperbola has a “center”, and two straight lines thatcradle the hyperbolas which are called asymptotes.
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HyperbolasA hyperbola has a “center”, and two straight lines thatcradle the hyperbolas which are called asymptotes.There are two vertices, one for each branch.
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HyperbolasA hyperbola has a “center”, and two straight lines thatcradle the hyperbolas which are called asymptotes.There are two vertices, one for each branch.The asymptotes are the diagonals of a box with the vertices ofthe hyperbola touching the box.
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HyperbolasA hyperbola has a “center”, and two straight lines thatcradle the hyperbolas which are called asymptotes.There are two vertices, one for each branch.The asymptotes are the diagonals of a box with the vertices ofthe hyperbola touching the box.
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HyperbolasThe center-box is defined by the x-radius a, and y-radius bas shown. b a
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HyperbolasThe center-box is defined by the x-radius a, and y-radius bas shown. Hence, to graph a hyperbola, we find the centerand the center-box first. b a
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HyperbolasThe center-box is defined by the x-radius a, and y-radius bas shown. Hence, to graph a hyperbola, we find the centerand the center-box first. Draw the diagonals of the boxwhich are the asymptotes. b a
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HyperbolasThe center-box is defined by the x-radius a, and y-radius bas shown. Hence, to graph a hyperbola, we find the centerand the center-box first. Draw the diagonals of the boxwhich are the asymptotes. Label the vertices and trace thehyperbola along the asympototes. b a
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HyperbolasThe center-box is defined by the x-radius a, and y-radius bas shown. Hence, to graph a hyperbola, we find the centerand the center-box first. Draw the diagonals of the boxwhich are the asymptotes. Label the vertices and trace thehyperbola along the asympototes. b aThe location of the center, the x-radius a, and y-radius b maybe obtained from the equation.
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HyperbolasThe equations of hyperbolas have the formAx2 + By2 + Cx + Dy = Ewhere A and B are opposite signs.
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HyperbolasThe equations of hyperbolas have the formAx2 + By2 + Cx + Dy = Ewhere A and B are opposite signs. By completing the square,they may be transformed to the standard forms below.
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HyperbolasThe equations of hyperbolas have the formAx2 + By2 + Cx + Dy = Ewhere A and B are opposite signs. By completing the square,they may be transformed to the standard forms below. (x – h)2 (y – k)2 (y – k)2 (x – h)2 a2 – b2 = 1 a2 = 1 – b2
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HyperbolasThe equations of hyperbolas have the formAx2 + By2 + Cx + Dy = Ewhere A and B are opposite signs. By completing the square,they may be transformed to the standard forms below. (x – h)2 (y – k)2 (y – k)2 (x – h)2 a2 – b2 = 1 a2 = 1 – b2 (h, k) is the center.
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HyperbolasThe equations of hyperbolas have the formAx2 + By2 + Cx + Dy = Ewhere A and B are opposite signs. By completing the square,they may be transformed to the standard forms below. (x – h)2 (y – k)2 (y – k)2 (x – h)2 a2 – b2 = 1 a2 = 1 – b2 x-rad = a, y-rad = b (h, k) is the center.
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HyperbolasThe equations of hyperbolas have the formAx2 + By2 + Cx + Dy = Ewhere A and B are opposite signs. By completing the square,they may be transformed to the standard forms below. (x – h)2 (y – k)2 (y – k)2 (x – h)2 a2 – b2 = 1 a2 = 1 – b2 x-rad = a, y-rad = b y-rad = b, x-rad = a (h, k) is the center.
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HyperbolasThe equations of hyperbolas have the formAx2 + By2 + Cx + Dy = Ewhere A and B are opposite signs. By completing the square,they may be transformed to the standard forms below. (x – h)2 (y – k)2 (y – k)2 (x – h)2 a2 – b2 = 1 a2 = 1 – b2 x-rad = a, y-rad = b y-rad = b, x-rad = a (h, k) is the center. Open in the x direction (h, k)
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HyperbolasThe equations of hyperbolas have the formAx2 + By2 + Cx + Dy = Ewhere A and B are opposite signs. By completing the square,they may be transformed to the standard forms below. (x – h)2 (y – k)2 (y – k)2 (x – h)2 a2 – b2 = 1 a2 = 1 – b2 x-rad = a, y-rad = b y-rad = b, x-rad = a (h, k) is the center. Open in the x direction Open in the y direction (h, k) (h, k)
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HyperbolasFollowing are the steps for graphing a hyperbola.
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HyperbolasFollowing are the steps for graphing a hyperbola.1. Put the equation into the standard form.
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HyperbolasFollowing are the steps for graphing a hyperbola.1. Put the equation into the standard form.2. Read off the center, the x-radius a, the y-radius b, and draw the center-box.
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HyperbolasFollowing are the steps for graphing a hyperbola.1. Put the equation into the standard form.2. Read off the center, the x-radius a, the y-radius b, and draw the center-box.3. Draw the diagonals of the box, which are the asymptotes.
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HyperbolasFollowing are the steps for graphing a hyperbola.1. Put the equation into the standard form.2. Read off the center, the x-radius a, the y-radius b, and draw the center-box.3. Draw the diagonals of the box, which are the asymptotes.4. Determine the direction of the hyperbolas and label the vertices of the hyperbola.
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HyperbolasFollowing are the steps for graphing a hyperbola.1. Put the equation into the standard form.2. Read off the center, the x-radius a, the y-radius b, and draw the center-box.3. Draw the diagonals of the box, which are the asymptotes.4. Determine the direction of the hyperbolas and label the vertices of the hyperbola. The vertices are the mid-points of the edges of the center-box.
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HyperbolasFollowing are the steps for graphing a hyperbola.1. Put the equation into the standard form.2. Read off the center, the x-radius a, the y-radius b, and draw the center-box.3. Draw the diagonals of the box, which are the asymptotes.4. Determine the direction of the hyperbolas and label the vertices of the hyperbola. The vertices are the mid-points of the edges of the center-box.5. Trace the hyperbola along the asymptotes.
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HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 42 22
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HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 42 22Center: (3, -1)
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HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 42 22Center: (3, -1)x-rad = 4y-rad = 2
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HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 42 22Center: (3, -1) 2x-rad = 4 4y-rad = 2 (3, -1)
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HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 42 22Center: (3, -1) 2x-rad = 4 4y-rad = 2 (3, -1)
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HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 4 2 22Center: (3, -1) 2x-rad = 4 4y-rad = 2 (3, -1)The hyperbola opensleft-rt
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HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 4 2 22Center: (3, -1) 2x-rad = 4 4y-rad = 2 (3, -1)The hyperbola opensleft-rt and the verticesare (7, -1), (-1, -1) .
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HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 4 2 22Center: (3, -1) 2x-rad = 4 (-1, -1) 4 (7, -1)y-rad = 2 (3, -1)The hyperbola opensleft-rt and the verticesare (7, -1), (-1, -1) .
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HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 4 2 22Center: (3, -1) 2x-rad = 4 (-1, -1) 4 (7, -1)y-rad = 2 (3, -1)The hyperbola opensleft-rt and the verticesare (7, -1), (-1, -1) .
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HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 4 2 22Center: (3, -1) 2x-rad = 4 (-1, -1) 4 (7, -1)y-rad = 2 (3, -1)The hyperbola opensleft-rt and the verticesare (7, -1), (-1, -1) .When we use completing the square to get to the standardform of the hyperbolas, because the signs, we add a numberand subtract a number from both sides.
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HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.
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HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:
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HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29
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HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29
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HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square
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HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x ) = 29
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HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29
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HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 16
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HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 16 –9
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HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9
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HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –94(y – 2)2 – 9(x + 1)2 = 36
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HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –94(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1
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HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –94(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 14(y – 2)2 – 9(x + 1)2 = 1 36 36
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HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –94(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 14(y – 2)2 – 9(x + 1)2 = 1 36 9 36 4
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HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –94(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 14(y – 2)2 – 9(x + 1)2 = 1 36 9 36 4 (y – 2)2 – (x + 1)2 = 1 32 22
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HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –94(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 14(y – 2)2 – 9(x + 1)2 = 1 36 9 36 4 (y – 2)2 – (x + 1)2 = 1 32 22Center: (-1, 2),
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HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –94(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 14(y – 2)2 – 9(x + 1)2 = 1 36 9 36 4 (y – 2)2 – (x + 1)2 = 1 32 22Center: (-1, 2), x-rad = 2, y-rad = 3
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HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –94(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 14(y – 2)2 – 9(x + 1)2 = 1 36 9 36 4 (y – 2)2 – (x + 1)2 = 1 32 22Center: (-1, 2), x-rad = 2, y-rad = 3The hyperbola opens up and down.
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HyperbolasCenter: (-1, 2),x-rad = 2, (-1, 5)y-rad = 3The hyperbola opens up and down.The vertices are (-1, -1) and (-1, 5). (-1, 2) (-1, -1)
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HyperbolasCenter: (-1, 2),x-rad = 2, (-1, 5)y-rad = 3The hyperbola opens up and down.The vertices are (-1, -1) and (-1, 5). (-1, 2) (-1, -1)
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HyperbolasExercise A. Write the equation of each hyperbola.1. (4, 2) 2. 3. (2, 4) (–6, –8)4. 5. 6. (5, 3) (2, 4) (–8,–6) (3, 1) (0,0) (2, 4)
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HyperbolasExercise B. Given the equations of the hyperbolasfind the center and radii. Draw and label the centerand the vertices. 7. 1 = x2 – y2 8. 16 = y2 – 4x2 9. 36 = 4y2 – 9x2 10. 100 = 4x2 – 25y211. 1 = (y – 2)2 – (x + 3)2 12. 16 = (x – 5)2 – 4(y + 7)213. 36 = 4(y – 8)2 – 9(x – 2)214. 100 = 4(x – 5)2 – 25(y + 5)215. 225 = 25(y + 1)2 – 9(x – 4)216. –100 = 4(y – 5)2 – 25(x + 3)2
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HyperbolasExercise C. Given the equations of the hyperbolasfind the center and radii. Draw and label the centerand the vertices.17. x –4y +8y = 5 2 2 18. x2–4y2+8x = 20 20. y –2x–x +4y = 6 2 219. 4x –y +8y = 52 2 221. x –16y +4y +16x = 16 2 2 22. 4x2–y2+8x–4y = 423. y +54x–9x –4y = 86 2 2 24. 4x2+18y–9y2–8x = 41
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