Conic Sections Review

Front

What is a hyperbola?

Back

A hyperbola is a type of conic section that appears as two separate curves, known as branches, which open away from each other. It is defined as the set of all points where the difference of the distances to two fixed points (foci) is constant.

Front

What is the standard form of a hyperbola with a horizontal transverse axis?

Back

The standard form is (x-h)²/a² - (y-k)²/b² = 1, where (h, k) is the center, a is the distance from the center to the vertices, and b is the distance related to the asymptotes.

Front

What are the asymptotes of a hyperbola?

Back

Asymptotes are lines that the hyperbola approaches but never touches. For a hyperbola in standard form, the equations of the asymptotes can be found using y = k ± (b/a)(x-h).

Front

How do you find the foci of a hyperbola?

Back

The foci of a hyperbola can be found using the formula c = √(a² + b²), where c is the distance from the center to each focus.

Front

What is a parabola?

Back

A parabola is a conic section that is defined as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).

Front

What is the standard form of a parabola that opens to the right?

Back

The standard form is (y-k)² = 4p(x-h), where (h, k) is the vertex and p is the distance from the vertex to the focus.

Front

What is the vertex of a parabola?

Back

The vertex is the point where the parabola changes direction. It is the highest or lowest point on the graph, depending on the orientation.

Front

What is the directrix of a parabola?

Back

The directrix is a fixed line used in the definition of a parabola. It is located p units away from the vertex in the opposite direction of the focus.

Front

What is the relationship between the coefficients in the equation of a hyperbola and its graph?

Back

The coefficients a² and b² in the hyperbola's equation determine the shape and orientation of the hyperbola. The larger the value of a², the wider the branches.

Front

How do you determine the orientation of a hyperbola?

Back

The orientation is determined by the placement of the x and y terms in the standard form. If the x-term is positive, the hyperbola opens left and right; if the y-term is positive, it opens up and down.