Hyperbolas are two curves that each have a fixed focus point and are mirror images of each other. They are similar to parabolas, and are shown on a graph in a similar fashion, except there are two instead of just one. A hyperbola, in most cases, appears to be an ellipsis that has been cut in half and reversed so the lines point outward instead of connecting together. The various attributes of hyperbolas make them unique and easy to distinguish from other types of graphs.
Basics of Hyperbolas
The hyperbola is made of two mirror-imaged parabolas. They do not ever touch the lines go in opposite directions. They also are open and do not ever end. This means they continue indefinitely and the same ratios and other attributes will still apply no matter how large the graph is drawn.
Center Point and Vertices
The center point of the hyperbola is not actually on either of the curves. Instead, it’s located between both of them, at equal distances from the vertices of each of the parabolas. The vertices are the point at which the curve actually turns and are the closest point on each curve to the center point.
Focus Point and Directrix
The focus point is a point inside of the curve. The focus point for both halves of the hyperbola is going to be an equal distance away from the center point. The directrix is a line next to the curve, close to the vertices, that can be used to describe the hyperbola. There is one directrix for each curve on the hyperbola, and they are the same for both halves.
Eccentricity of the Curve
The eccentricity is how much the curve differs from a circle. It is a ratio made of the distance to the line from the focus point and the directrix. For all hyperbolas, the eccentricity is going to be greater than 1. The eccentricity can be found by using the formula √(a^2+b^2)/a. The ratio is going to be the same no matter which point on the curve is used.
Symmetry
Each hyperbola has a directrix that can be used as part of the ratio for the eccentricity. Perpendicular to this line is the axis of symmetry, which includes both focus points. This line is where the curve turns and begins going in the opposite direction. It passes through the vertices of both halves of the hyperbola. Both sides of the axis of symmetry are exactly the same.
Latus Rectum
This is a line that is parallel to the directrix and goes through the focus point. It can be determined using the equation 2b^2/a and is always going to be the same for both parts of the hyperbola.
Hyperbolas can appear to be complex, but there are a variety of attributes that apply to every one. This makes it easier to solve a hyperbola and to graph it. Understanding each of the attributes can help the student learn how to use these to solve various math problems and to understand what is happening when they see a hyperbola on a graph.