Problem 84 - Precalculus with Limits

Question

TRUE OR FALSE? In Exercises 83 and 84, 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 true.

Step 1: Understand the definition of a parabola

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). This definition underlies the properties of a parabola, including the positioning of the focus and directrix.

Step 2: Apply the definition to the statement

If a parabola's vertex and focus lie on a horizontal line, this indicates that the parabola is opening either to the right or to the left. In both these cases, the directrix of the parabola, in order to be equidistant from the focus at any point on the parabola, should be vertical.

Step 3: Validate the statement

According to the definition of a parabola, if the vertex and focus of a parabola lie on a horizontal line, then the directrix is vertical. This corresponds to the given statement, hence confirming it is true.

Key Concepts

Vertex

The vertex of a parabola is a crucial point where the shape changes direction. It acts as the midpoint between the focus and the directrix. This makes it a key element in defining the parabola’s structure.

• The vertex lies directly on the axis of symmetry of the parabola.
• For a parabola that opens horizontally, the vertex is where the parabola turns left or right.

Understanding the position of the vertex is essential because it influences the orientation and direction of the parabola. It's like the hinge that determines how the parabola opens, offering a visual clue to the parabola's path.

Focus

The focus is a fixed point that, along with the directrix, helps construct a parabola. It’s one of the defining features because every point on the parabola is equidistant to this point and the directrix.

• If the parabola opens horizontally, the focus lies to the left or right of the vertex on the same horizontal line.
• The distance between the vertex and the focus helps define the width and steepness of the parabola.

Knowing the location of the focus is vital for graphing the parabola accurately. It allows you to understand how wide or narrow the parabola will appear, depending on its distance from the vertex.

Directrix

The directrix of a parabola is a fixed line used in conjunction with the focus to define the shape. It serves as a boundary, determining the set of equidistant points that form the parabola.

• For a horizontally opening parabola, the directrix is a vertical line.
• This line is opposite the focus with the vertex in the middle.

The directrix helps in plotting the parabola by establishing a reference point opposite the focus. Knowing it assists students in understanding how to construct the parabola's shape from these geometric constraints.

Horizontal Line

A horizontal line in the context of parabolas often refers to the alignment of key features like the vertex and the focus. When both of these lie on the same horizontal line, it signifies how the parabola is oriented.

• This arrangement means the parabola opens either to the left or right, not up or down.
• The horizontal alignment simplifies understanding because one can predict the direction of opening based on the focus’s side.

Visualizing these elements on a horizontal line provides clarity in analyzing and graphing parabolas. It lays the foundation for predicting the parabola’s direction and understanding its geometric positioning.