Problem 61 - Algebra 2

Question

The directrix of a parabola is the line \(y=-2 .\) The focus is the point \((0,2)\) a. Suppose the directrix remains fixed, but the focus is shifted along the \(y\) -axis farther away from the directrix. Explain what happens to the vertex and the shape of the parabola. b. Suppose the focus moves along the \(y\) -axis toward the directrix. Explain what happens the focus moves along the \(y\) -axis toward the directrix. Explain what happens to the vertex and the shape of the parabola. c. What would happen if the focus moved down all the way to the directrix?

Step-by-Step Solution

Answer

a. As the focus moves away from the directrix, the vertex shifts upwards and the parabola narrows. b. As the focus moves towards the directrix, the vertex moves downwards and the parabola widens. c. If the focus moves down all the way to the directrix, the parabola becomes a straight line as the vertex coincides with the directrix.

Step 1: Analysing the Geometry and Positioning

The position or location of the vertex of a parabola lies midway between the focus and the directrix. The axis of symmetry of the parabola is the line through the focus perpendicular to the directrix.

Step 2: Effects of Moving the Focus Farther Away from the Directrix

As the focus moves away from the directrix, the vertex also shifts upwards along the \(y\)-axis. This is because the vertex is always midway between the focus and the directrix. The parabola will become narrower since the distance between the focus and the directrix is increasing.

Step 3: Effects of Moving the Focus towards the Directrix

If the focus moves along the \(y\)-axis towards the directrix, the vertex will also move downwards along the \(y\)-axis. The parabola will become wider since the distance between the focus and the directrix is decreasing.

Step 4: Position of the Focus Mirroring the Directrix

If the focus moves down all the way to the directrix, technically, the parabola would become a straight line because the vertex (that lies midway) would coincide with the directrix.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a special point located exactly midway between the focus and the directrix of the parabola. Think of it as the balancing point of the parabola's curve. If you draw a vertical line connecting the focus to the directrix, the vertex will lie at the midpoint of this line.

• When the focus is closer to the directrix, the vertex sits lower on the graph, leading to a wider parabola.
• Conversely, if the focus is farther from the directrix, the vertex shifts up and the parabola becomes narrower.
• If the focus and the directrix coincide, the vertex is on the directrix, resulting in a line.

Understanding the vertex helps us analyze the shape and position of the parabola on a graph, as it plays a central role in determining the parabola's symmetry and direction.

Focus and Directrix

A crucial aspect of a parabola is its focus and directrix. The focus is a fixed point, while the directrix is a fixed line. Both elements are essential in defining how the parabola is shaped.

• The parabola consists of points equidistant from both the focus and the directrix.
• If the focus is moved farther from the directrix, the parabola narrows down and the distance from the focus increases.
• If the focus approaches the directrix, the parabola widens, adapting to the changing distances.
• When the focus and directrix align, the parabola ceases to exist as it becomes a straight line.

Each adjustment in the focus or the directrix alters the entire nature of the parabola, highlighting their significance in the parabola's geometry.

Axis of Symmetry

The axis of symmetry is an imaginary vertical line that divides the parabola into two mirror-image halves. For any point on one side of this axis, there is an identical point on the opposite side, reflected across this line.

• It passes through the vertex and the focus.
• This means every parabola has a unique axis of symmetry.
• The equation of the axis of symmetry is derived from the vertex's coordinate, typically written as "x = h" for vertical parabolas.

Having this symmetrical property allows us to predict the parabola's behavior on either side of this axis, providing a clearer understanding of its geometric nature.