Which of the following is the equation of the hyperbola graphed?

A. (x + 5)2 9 − y − 32 9 = 1
Incorrect. In this equation, h = -5 and k = 3.

B. (x − 5)2 9 − y + 32 9 = 1
Correct! h = 5, k = -3, center (5, -3) x-radius = 3, y-radius = 3 and x term positive, so hyperbola opens in the “x-direction”.

C. (x − 5)2 3 − y + 32 3 = 1
Incorrect. Values in denominators should be x- and y-radii squared.

D. (x + 5)2 3 − y − 32 3 = 1
Incorrect. Check values of h and k. Values in denominators should be x-and y-radii squared.

Which of the following is the graph of (x − 3)2 25 − y2 16 = 1?

A. B.
C. D.

A
Correct! Center at (3, 0); x-radius = 5, y-radius = 4 hyperbola opens horizontally (in x-direction)

B
Incorrect. Check the x- and y-radius values.

C
Incorrect. Check the values of h and k—the center point.

D
Incorrect. Check the direction the hyperbola should open.

Which of the following best describes the differences in the graphs of the following two hyperbolas?

  1. (x − 7)2 16 − (y + 3)2 36 = 1
  2. (x − 7)2 36 − (y + 3)2 16 = 1
  1. Hyperbola I and II have the same radii, with center points (7, -3) and (-3, 7) respectively. They open vertically because the y-radius is longer in both.
  2. The two hyperbolas have the same center point (7, -3), but open in different directions because the positive terms are different.
  3. Hyperbola II is a translation of hyperbola I that has been moved 3 units to the left and 7 units up.
  4. Hyperbola I opens horizontally while hyperbola II opens vertically. The coordinates of the center points (based on h and k) are (7, -3) versus (-3, 7). The x-radii and y-radii are the same for the two hyperbolas.

A
Incorrect. The longest radius doesn't determine the direction a hyperbola opens.

B
Incorrect. Check the values of h and k for the center points.

C
Incorrect. The hyperbolas open in different directions and the changes in the center points are not as described here.

D
Correct! The direction a hyperbola opens depends on which term is positive in the equation. The values of h and k indicate the center point.

Which of the following pairs could be coordinates of the focal points of this hyperbola?

x2 36 − y2 64 = 1

A. (0, -10) and (0, 10)
Incorrect. Check the direction the hyperbola opens. The focal radius is 10, but location of points is incorrect.

B. (-10, 0) and (10, 0)
Correct! See the Resource section for a more detailed explanation.

C. (0, -5.29) and (0, 5.29)
Incorrect.

D. (-5.29, 0) and (5.29, 0)
Incorrect.

Which of the following is a hyperbola which you would graph with center at (-4, 5), x-radius length 6 units and y-radius length 5 units and opening vertically (in the y-direction)?

A. (y + 5)2 25 − x − 42 36 = 1
Incorrect. Check the values of h and k.

B. (y + 4)2 36 − y − 52 25 = 1
Incorrect. This one opens in the horizontally (in the x-direction) and the x- and y-radii are reversed.

C. (y − 5)2 25 − x + 42 36 = 1
Correct! h = -4, k = 5, x-radius= 6, y=radius = 5 and hyperbola opens vertically (y-term positive)

D. (y − 5)2 36 − x + 42 25 = 1
Incorrect. x-and y-radii are reversed.