Hyperbolas By: Jillian Venditti, Gabriella Colaiacovo, Jessica Spadaccini, and Giuliana Izzo

 Hyperbola- is a set of all points P in the Plane, the difference of whose distances from 2 fixed points called the foci, is a constant.  The line through the foci is known as the Foci Axis.  The point that is midway between the foci is called the Center.  The points where the hyperbola intersects its focal axis are the Vertices.  Transverse axis- line containing the foci.  Conjugate Axis- line perpendicular to the transverse axis at the center.  Equidistant- is the same distance from a point to another point. Key Terms

 Hyperbola symbols When the transverse axis is horizontal: a- the distance from the center to the vertex b- the height of the rectangle c- the distance from the center to the foci Center is always (h,k) Equation: Horizontal

 When the transverse axis is vertical: a- the width of the rectangle b- the distance from the center to the vertex c- the distance from the center to the foci Center is always (h,k) Equation: Vertical

 Steps for Solving/Graphing a Hyperbola when Given a Point(s): 1. Graph the given points- whether given vertices, foci, or the center. 2. Find the remaining points:  The center is equidistant from the two vertices.  Both foci have the same distance to the center.  Both vertical vertices have the same distance to the center.  Both horizontal vertices have the same distance to the center. Also: a- Represents the distance from the center to each horizontal vertice b- Represents the distance from the center to each vertical vertice c- Represents the distance from the center to each focus (You may use the Pythagorean Theorem: a² + b² = c² to solve for these values and graph your points.) Step by Step for Solving Equations for Hyperbola

3. Once you have graphed all foci, vertices, and the center, connect all vertices by drawing a dotted square box. 4. Solve for the asymptotes. The general equation is: (y-k) = (+/-) b (x-h) a 5. Graph your asymptotes. (They should go through the corners of your box.) Con’t of Steps

6. Draw the hyperbola using your vertices and your asymptotes.  The transverse axis of a horizontal hyperbola is parallel to the x- axis while the transverse axis of a vertical hyperbola is parallel to the y-axis. 7. Find the equation for the hyperbola using Center (h,k). Use the following equations: Horizontal: (x-h) ² - (y-k) ² = 1 a² b² Vertical: (y-k) ² - (x-h) ² = 1 b² a² Con’t of Steps

1.Analyze the given equation: group your x’s and y’s on one side and integers on the other. 2.Factor. 3.Add (1/2 b) ² to your factored binomial to create a polynomial. Distribute to get your values for the other side. 4.Factor your polynomials completely and divide to get your equation. Steps For Solving Hyperbolas Using Completing the Square to State Key Points and Asymptotes

5. Use your equation to get the values for a, b, and c. Equations: Horizontal: (x-h)² - (y-k)² = 1 a² b² Vertical: (y-k)² - (x-h)² = 1 b² a² a- Represents the distance from the center to each horizontal vertice. b- Represents the distance from the center to each vertical vertice. c- Represents the distance from the center to each focus. Con’t of Steps

Use the Pythagorean Theorem to solve for c. 6. Using the Center, you can find the values for the vertices and foci. If: Horizontal- change in x Vertical- change in y 7. Find your asymptotes using the equation: (y-k) = (+/-) b (x-h) a Con’t of Steps

 Given: F₁ (-4,0), V₂ (-4,4), V₁ (-4,2)  Graph and State all Points (Center, Foci, Vertices, a, b and c) also find the Equation for the Asymptotes. Examples:

Use the Step by Step Slide to solve the problem: Step 1: Red Step 2: Brown Step 3: Purple Step 4: Green Step 5: Orange

 Analyze the following Hyperbola using completing the square. State all points (Center, Foci, Vertices, a, b and c) also find the Equation for the Asymptotes.  Equation: 2x²- y²+ 2y + 8x + 3 = 0

Use Step by Step Slide to solve the hyperbola equation: Step 1: Red Step 5:Purple Step 2: Brown Step 6: Green Step 3: Blue Step 7: Pink Step 4: Yellow

 e/Precalculus/Lectures/Hyperbolas e/Precalculus/Lectures/Hyperbolas  erials/Lecture%20Notes/PreCalculus%20I/Chapter%209/ 9.2.pdf erials/Lecture%20Notes/PreCalculus%20I/Chapter%209/ 9.2.pdf References