Parabola

A parabola is a U-shaped curve that can be defined as the set of all points that are equidistant from a fixed line called the directrix and a fixed point called the focus. The standard form of a parabola is given by the equation y = ax^2 + bx + c or x = ay^2 + by + c.

Key Concepts

• Vertex: The turning point of the parabola, either the highest point (in the case of a downward-opening parabola) or the lowest point (in the case of an upward-opening parabola).
• Axis of Symmetry: The vertical line that passes through the vertex of the parabola and divides it into two symmetrical halves.
• Directrix: The fixed line that is equidistant from all points on the parabola.
• Focus: The fixed point that is equidistant from all points on the parabola.
• Standard Form: The general form of the equation of a parabola, either in terms of x or y.

Equations

The standard form of a parabola can be written in two different ways:

1. For a parabola opening vertically: y = ax^2 + bx + c
2. For a parabola opening horizontally: x = ay^2 + by + c

Graphing a Parabola

To graph a parabola, use the following steps:

1. Identify the vertex, axis of symmetry, and direction of opening.
2. Plot the vertex on the coordinate plane.
3. Use the axis of symmetry to plot additional points symmetrically on either side of the vertex.
4. Draw the parabolic curve through the plotted points.

Applications

Parabolas have numerous real-world applications, including in physics (e.g., projectile motion), engineering (e.g., designing arches), and astronomy (e.g., orbits of celestial bodies).

Practice Problems

1. Find the vertex, axis of symmetry, and direction of opening for the parabola y = 2x^2 - 4x + 1.

2. Graph the parabola x = 3y^2 - 6y + 2.

3. A ball is thrown into the air with an initial vertical velocity of 20 m/s. The height of the ball at time t seconds is given by the equation h = -5t^2 + 20t + 10. What is the maximum height reached by the ball?

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