11'5 Translation of Conics

1
11.5 Translation of Conics

• In previous sections, all conic sections had
  centers at (0,0).
• In this section the conic sections can have any
  center with ordered pair (h,k).
• For the parabola, the vertex will be (h,k).

2
Standard Equations
3
Graphing a Translated Circle

• Graph the circle (x 1)2 (y 2)2 9
• Procedure
• From the standard equation, the center is (h,k)
• To correctly derive (h,k) from the equation,
  change the signs.
• Therefore, h 1 and k -2 giving a center
  (1,-2)
• The radius is sqrt(9) 3. Therefore, r 3
• Plot the center (1,-2), and lightly draw a new
  axis through the pt.
• The new axis should match up with the original
  x-y axis.
• Put a point along each axis r distance, or 3,
  from the center.
• Sketch the circle through the four points.

4
Graphing a Translated Ellipse

• Graph the ellipse (x 2)2/9 (y 1)2/4 1
• Procedure
• From the standard equation, changing signs, h
  2, k 1.
• The largest number is under x, so the ellipse is
  horizontal.
• a 3 and b 2
• You do not need to graph the foci, unless asked
  to.
• As with the circle, plot the center (2,1), and
  draw a new axis through the point.
• Plot a and a along the new x-axis.
• Plot b and b along the new y-axis.
• Carefully sketch the ellipse through the plotted
  points.

5
Graphing a Translated Hyperbola

• Graph the hyperbola (x 3)2/1 (y 2)2/9 1
• Procedure
• From the standard equation, changing signs, h
  3 and k 2.
• The hyperbola is horizontal because the negative
  is in front of y.
• a 1 and b 3 and center (3,2)
• As previously done, plot the center point and
  draw a new axis through this point.
• Plot a and a along the new x-axis, and b and
  b along the new y-axis.
• Draw the box with dashed lines through a and b.
• Lightly draw the asymptotes as diagonal lines
  through the corners of the box.
• Sketch the hyperbola through point as using the
  asymptotes as a guide.

6
Graphing a Translated Parabola

• Graph the parabola (x 2)2 4(-1)(y 3)
• Procedure
• From the standard equation, changing signs, h
  2 and k 3.
• The vertex of the parabola is (2,3)
• The parabola is vertical since the x is squared
  and p -1
• Since p lt 0, the parabola opens down.
• As was previously done, plot the center, (the
  vertex), and draw a new axis through the point.
• Since p -1, plot the point p one place down
  from the vertex.
• Sketch the parabola through the vertex facing
  down.