Problem 78 - Precalculus

Question

Determine whether the statement is true or false. Justify your answer. If the vertex and focus of a parabola are on a horizontal line, then the directrix of the parabola is vertical.

Step-by-Step Solution

Answer

The statement is false. When the vertex and focus of a parabola are on a horizontal line, the directrix of the parabola is a horizontal line, not a vertical one.

Step 1: Understand parabola properties

A parabola is defined as the set of all points in a plane that are equidistant from a fixed point (the focus), and a fixed line (the directrix).

Step 2: Recognize relationship between vertex, focus, and directrix

The vertex of the parabola is the point exactly halfway between the focus and the directrix. The axis of the parabola is the straight line passing through the focus and the vertex.

Step 3: Reflect on statement

Consider the configuration where the vertex and the focus of the parabola are located on a horizontal line. This positions the parabola in an 'upward' or 'downward' orientation, i.e., the parabola opens either upwards or downwards.

Step 4: Consider directrix orientation

With the parabola oriented upwards or downwards, the directrix must be a horizontal line, lying above or below the focus respectively. The axis (line through the focus and the vertex) in this case will also be vertical.

Step 5: Confirm statement

Since the directrix is horizontal when the vertex and focus are on a horizontal line (not vertical as the statement suggests), the original statement is false.

Key Concepts

Vertex

The vertex is a central concept when dealing with parabolas. It is the point where the parabola makes its sharpest turn and can be thought of as the "peak" or "lowest point" depending on its orientation. The vertex serves as a sort of "hinge" for the parabola. It is crucial because it lies exactly halfway between the focus, which is a fixed point, and the directrix, which is a fixed line. This makes it a midpoint that defines the precise structure and symmetry of the parabola.

• The vertex is always equidistant from the focus and the directrix.
• It is the location where the parabola intersects its axis of symmetry.

Understanding the vertex helps to swiftly determine the direction in which a parabola opens (upward, downward, left, or right), and this can be especially useful when graphing or solving equations involving parabolas.

Focus

The focus of a parabola is a fixed point that, along with the directrix, defines the shape of the parabola. The parabola consists of all points that are equidistant from the focus and the directrix. This unique geometric property causes light or sound waves to converge to or emanate from the focus, which is why parabolic shapes are often used in satellite dishes and headlights.

• The position of the focus influences whether the parabola opens upward, downward, sideways left, or sideways right.
• It is located on the parabolic curve’s axis of symmetry.

Knowing the position of the focus is essential in applications where precise pathways or signals are needed.

Directrix

The directrix is a straight line that, in tandem with the focus, defines a parabola. It is crucial in the geometric definition of a parabola, as every point on the parabola is equidistant from both the directrix and the focus.

• When the focus and vertex are on a horizontal line, the directrix is actually a vertical line, contrary to some misconceptions.
• The directrix never intersects with the parabola; instead, it helps in maintaining its symmetry and shape.

Understanding the orientation of the directrix is key in determining how a parabola is structured. It is also helpful for identifying the position and magnitude of the parabola's vertex.

Axis of Symmetry

The axis of symmetry is a line that runs through the vertex and the focus, perfectly dividing the parabola into two symmetric halves. This line is essential for understanding the orientation and balance of the parabola, acting as a mirror where one side is the exact reflection of the other.

• The axis of symmetry is perpendicular to the directrix.
• For vertical parabolas, it is a vertical line and for horizontal parabolas, it is a horizontal line.

This property is particularly valuable when solving quadratic equations or graphing parabolas, as it simplifies the identification of points and other characteristics. Knowing the axis of symmetry can help predict the behavior of the curve in geometric constructions.