1) Describe the similarities and differences between hyperbolas and ellipses. 2) List the properties of a hyperbola that allow you to sketch its graph.
1. An ellipse is a closed figure, while a parabola is not.
1) Similarities between hyperbolas and ellipses:
- Both hyperbolas and ellipses are conic sections, meaning they are formed by the intersection of a plane and a cone.
- They both have two branches or components.
Differences between hyperbolas and ellipses:
- In an ellipse, the sum of the distances from any point on the ellipse to two fixed points called foci is constant. In a hyperbola, the difference of the distances from any point on the hyperbola to two fixed points called foci is constant.
- The shape of an ellipse is generally more circular, while a hyperbola has more elongated, open branches.
- The equation of an ellipse is generally in standard form: (x-h)^2/a^2 + (y-k)^2/b^2 = 1 or (x-h)^2/b^2 + (y-k)^2/a^2 = 1, while the equation of a hyperbola is in the form: (x-h)^2/a^2 - (y-k)^2/b^2 = 1 or (y-k)^2/b^2 - (x-h)^2/a^2 = 1, where (h, k) represents the center of the conic section.
2) Properties of a hyperbola that allow you to sketch its graph:
- Center: The coordinates of the center (h, k) will help determine the position of the hyperbola on the coordinate plane.
- Vertices: The vertices are the points on the transverse axis of the hyperbola and are located a distance of 'a' units from the center.
- Foci: The foci are the two fixed points inside the hyperbola that determine its shape. The distance between the foci and the center is denoted by 'c' and can be calculated using the equation: c^2 = a^2 + b^2.
- Asymptotes: The asymptotes are imaginary lines that serve as boundaries for the hyperbola. They have slopes of ±(b/a) and pass through the center of the hyperbola.
- Transverse axis: The transverse axis is the line segment passing through the center and perpendicular to the asymptotes. It connects the vertices of the hyperbola.
- Conjugate axis: The conjugate axis is the line segment passing through the center and perpendicular to the transverse axis.
- Eccentricity: The eccentricity of the hyperbola, denoted by 'e', determines how stretched or compressed the hyperbola is. It can be calculated using the formula: e = c/a.
1) To describe the similarities and differences between hyperbolas and ellipses, we need to understand their definitions and key properties.
Similarities:
- Both hyperbolas and ellipses are conic sections, which means they are formed by intersecting a cone with a plane.
- Both curves have two foci, represented as points inside the curve, and the sum of the distances from any point on the curve to the foci is constant.
- The shape of both curves is determined by two parameters: the distance between the foci and a constant value called eccentricity.
Differences:
- Ellipses have eccentricities between 0 and 1, while hyperbolas have eccentricities greater than 1.
- The major and minor axes of an ellipse are perpendicular, whereas the transverse and conjugate axes of a hyperbola are perpendicular.
- The curves also have different equations. The general equation of an ellipse is (x^2 / a^2) + (y^2 / b^2) = 1, while the general equation of a hyperbola is (x^2 / a^2) - (y^2 / b^2) = 1.
2) To sketch the graph of a hyperbola, you need to know its key properties:
a) Center: The center of a hyperbola is the midpoint between the foci. It is represented by the coordinates (h, k).
b) Asymptotes: Hyperbolas have two asymptotes that pass through the center. The equations of the asymptotes are y = k ± (b / a)(x - h), where (h, k) is the center, and a and b are the lengths of the semi-major and semi-minor axes, respectively.
c) Vertices: The vertices of a hyperbola are the points where the curve intersects its transverse axis. They are located a distance a units to the left and right of the center, forming a rectangle with the foci as diagonals.
d) Foci: Hyperbolas have two foci, represented by points (h ± c, k), where c is the distance between the center and a focus. The value of c can be calculated using the equation c = √(a^2 + b^2).
e) Asymptote Intercepts: The asymptotes intersect the coordinate axes at points that can be calculated using the values of a and b.
Once you have these properties, you can plot the center, vertices, foci, asymptotes, and asymptote intercepts on a coordinate system to get a rough sketch of the hyperbola.