A hyperbola is formed when the two cones are pointing towards one another, but not touching, forming a mirror image of one another. As a result, the figure looks like two disconnected curves that are opposite one another. The curves of this figure are often referred to as arms or branches. The points of the curves that are closest together are called the vertices. Never did understanding this shape become more vital then when we travel into space. The trajectory of objects that are acted upon by gravity is tracked using this figure. Large bodies that exert gravity on objects in space will pull it towards it until the object reaches escape velocity. As an object escapes, it forms a hyperbolic shape for the path that it travels. This series of worksheets and lessons will help students learn to write and understand equations for hyperbolas.
Aligned Standard: High School Geometry - HSG-GPE.A.3
- Finding Hyperbolas Step-by-step Lesson - What is the equation based on the center, vertex, and focus?
- Guided Lesson - Question number three mixes it up a bit to make sure you are reading everything.
- Guided Lesson Explanation - There is a general formula listed on the first page that applies to all questions this way I didn't need to write it over and over.
- Practice Worksheet - Go hyperbola equation crazy with these.
- Matching Worksheet - We play equation tennis. Go back and forth until you find matches for all your work.
- Hyperbolas - Find the Foci and Vertices Worksheet Five Pack - Some of these equations are bit difficult to interpret.
- Hyperbolas Equations Worksheet Five Pack - More work like the previous pack.
- Answer Keys - These are for all the unlocked materials above.
Given a few measures and location, we want you to find the equation of the hyperbolas for us.
- Homework 1 - The point that the hyperbola is focused (pointed) on is referred to as the center.
- Homework 2 - When we use a coordinate system, the recognizable point that is on the branch closest to the center of the hyperbola is called the vertex.
- Homework 3 - The foci reside inside each branch of the hyperbola.
These sheets get progressively harder.
- Practice 1 - Find an equation of the hyperbola with x-intercepts at x = –12 and x =6, and foci at (–16, 0) and (10, 0).
- Practice 2 - If we look at the foci, we will see that they are side-by-side. This indicates that branches of the hyperbola follow this lead. This also means that the center, foci, and vertices are on a line that is parallel to the x-axis.
- Practice 3 - The center resides on the x-axis. This means that the xintercepts have to also be vertices for the hyperbola.
Math Skill Quizzes
If you handled the homework and practice in stride, these are pretty straight forward for you.
- Quiz 1 - Find an equation for the hyperbola with the center at (4, 14), vertex at (0, 14), and focus at (18, 14).
- Quiz 2 - The center is in the middle of the foci.
- Quiz 3 - Write the standard equation of each Hyperbola: x2 / 9 – (y+4)2 / 36 = 1
How to Find the Equation of a Hyperbola?
In analytic geometry, there are several different types of curves, and the most common ones include hyperbolas and ellipses. A hyperbola is a smooth curve lying in a plane. When a right circular cone intersects with a plane at a specified angle, which cuts both halves of the cone, it produces two unbounded curves, which we call the hyperbolas. Both curves are a reflection of each other and are not connected.
A hyperbola may occur vertically or horizontally; horizontal: (x-h)2/a2 - (y-k)2/b2 =1, vertical: (y-k) 2/a2 -(x-h)2/b2 = 1. These are the two patterns of a hyperbola, and you can determine the equation of a hyperbola using these patterns. These are used when the center is not the origin. When the origin is the center, the pattern becomes; horizontal: x2/a2 -y2/b2 =1, vertical: y2/a2 -x2/b2 = 1. Here, (h,k) are the coordinates of the center point. (±a,0) are the vertices. (0,±b) are the coordinates of co-vertices. a and b are connected via the formula, c2 = a2+b2 (±c 0) are the coordinates of the foci. The distance between two vertices is equal to 2a. The center can be calculated using the midpoint formula using the two vertices. Midpoint=((x1+x2) / 2 ,(y1 + y2)/2) Distance between the center and the focus is given by ae. e here is the eccentricity. By substituting all these values in the equation, you can get the equation of a hyperbola.