Focus, Vertex and Directrix of Parabolas

Front

What is the definition of a parabola?

Back

A parabola is a symmetric curve formed by the intersection of a cone with a plane parallel to its side. It can be defined as the set of all points equidistant from a fixed point (focus) and a fixed line (directrix).

Front

What are the coordinates of the focus in the equation (y - k)^2 = 4p(x - h)?

Back

The coordinates of the focus are (h + p, k) for a right-opening parabola and (h - p, k) for a left-opening parabola.

Front

What is the directrix of a parabola given by the equation (y - k)^2 = 4p(x - h)?

Back

The directrix is the line x = h - p for a right-opening parabola and x = h + p for a left-opening parabola.

Front

What does the parameter 'p' represent in the parabola equations?

Back

The parameter 'p' represents the distance from the vertex to the focus and the distance from the vertex to the directrix.

Front

How do you determine the direction a parabola opens based on its equation?

Back

If the x-term is squared (e.g., (y - k)^2 = 4p(x - h)), the parabola opens left or right. If the y-term is squared (e.g., (x - h)^2 = 4p(y - k)), it opens up or down.

Front

What is the vertex of a parabola given by the equation (y - k)^2 = 4p(x - h)?

Back

The vertex of the parabola is at the point (h, k).

Front

For the equation (y - 4)^2 = -8(x + 1), what is the value of 'p'?

Back

The value of 'p' is -2.

Front

In the equation y^2 = 4px, what does '4p' represent?

Back

'4p' is the coefficient that determines the width of the parabola; larger values of '4p' result in a wider parabola.

Front

What is the focus of the parabola given by the equation y^2 = -8x?

Back

The focus is at the point (-2, 0).

Front

How can you identify a left-opening parabola from its equation?

Back

A left-opening parabola has a negative value for 'p' in the standard form (y - k)^2 = 4p(x - h).