How to Graph Hyperbolas and Find Their Key Features

TL;DR

To graph hyperbolas, use the formulas: horizontal hyperbolas are represented by x²/a² - y²/b² = 1, while vertical hyperbolas use y²/a² - x²/b² = 1. Identify the center, vertices at (+/-a, 0) or (0, +/-a), and foci at (+/-c, 0) or (0, +/-c) where c = √(a² + b²). Asymptote equations are y = +/- (b/a)x for horizontal and y = +/- (a/b)x for vertical hyperbolas.

Transcript

in this video we're going to focus on hyperbolas now you may want to get a sheet of paper and a pen to write down some notes so we have a horizontal hyperbola on the left and a vertical hyperbola on the right the formula that corresponds to the horizontal hyperbola is x squared over a squared minus y squared over b squared which is equal to one thi... Read More

Key Insights

• 🚥 The formula for horizontal hyperbolas is x^2/a^2 - y^2/b^2 = 1, while the formula for vertical hyperbolas is y^2/a^2 - x^2/b^2 = 1.
• 😃 The vertices and foci of a hyperbola are determined by the values of a and b, with the foci located c units away from the center.
• 🚥 The equations of the asymptotes depend on whether the hyperbola is horizontal or vertical, and whether it is centered at the origin or shifted.
• #️⃣ The domain of a hyperbola is all real numbers, while the range is determined by the y-values of the vertices and any breaks in the hyperbola's curve.
• ↔️ Hyperbolas can open left and right or up and down, with their shape determined by the signs and placement of the variables in the equation.

Questions & Answers

Q: How do you graph a hyperbola?

To graph a hyperbola, first find the center, vertices, and foci. Then draw a rectangle connecting the vertices and draw the asymptotes diagonally through the center. Finally, sketch the curve that follows the asymptotes.

Q: How do you find the coordinates of the vertices?

For horizontal hyperbolas centered at the origin, the vertices have coordinates (+/-a, 0). For vertical hyperbolas centered at the origin, the vertices have coordinates (0, +/-a). If the center is not at the origin, add the coordinates of the center to the vertices.

Q: How do you find the coordinates of the foci?

For both horizontal and vertical hyperbolas, the coordinates of the foci are (+/-c, 0) or (0, +/-c), where c = √(a^2 + b^2). If the center is not at the origin, add the coordinates of the center to the foci.

Q: What are the equations of the asymptotes?

For horizontal hyperbolas centered at the origin, the equation of the asymptotes is y = +/- (b/a) * x. For vertical hyperbolas centered at the origin, the equation of the asymptotes is y = +/- (a/b) * x. If the center is not at the origin, adjust the equations by subtracting the coordinates of the center.

Summary & Key Takeaways

• The formula for horizontal hyperbolas is x^2/a^2 - y^2/b^2 = 1, and for vertical hyperbolas is y^2/a^2 - x^2/b^2 = 1.

• The vertices for horizontal hyperbolas are (+/-a, 0) and for vertical hyperbolas are (0, +/-a).

• The foci for both types of hyperbolas are (+/-c, 0) or (0, +/-c), where c = √(a^2 + b^2).

• The equations of the asymptotes for horizontal hyperbolas are y = +/- (b/a) * x, and for vertical hyperbolas are y = +/- (a/b) * x.