Open the “Hyperbola” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Interactive Figures) or at bit.ly/3raFUGB.

(a) Set the slider to a = 2. Grab the point F₁, which represents a focus of a hyperbola, and move the point F₁ to (−2, 1). Grab the point F₂ , which represents the other focus of a hyperbola, and move the point F2 to (4, −3). What is the center of the hyperbola? Write your answer as an ordered pair. Confirm the center of the hyperbola is at the midpoint of the two foci.

(b) Click “Show Trace.” Grab Point A and move it around the Cartesian plane. Complete a trace of the entire hyperbola (be sure your hyperbola has two branches). What is the absolute value of the difference of the distances from the foci to any point on the hyperbola?

(c) Hit the reset button on the interactive figure (upper-right corner). Move the point F₁ to (−4, 0) and move the point F₂ to (4, 0). Click “Show Hyperbola.” Use the a slider to change the value of a from 2 to 3. What happens to the vertices as a changes? What does the value of a represent? Notice that the absolute value of the difference of the distances from the foci to any point on the hyperbola is 2a. Click “Show Trace” to verify this.

(d) What is the value of c, the distance from the center to either focus?

(e) Check the box “Show Equation of Hyperbola.” What is the equation of the hyperbola?

(f) Hit the reset button on the interactive figure (upper-right corner). Move the point F₁ to (0, −5) and move the point F₂ to (0, 5). Set the value of a to 3. Check the box “Show Hyperbola.” What is the value of c, the distance from the center to either focus?

(g) Click “Show Equation of Hyperbola.” What is the equation of the hyperbola?