Parabola

Definition

A parabola is a U-shaped curve that is the graph of a quadratic function. It is one of the four fundamental conic sections, which are the shapes formed by the intersection of a plane and a cone.

5 Must Know Facts For Your Next Test

1. The equation of a parabola in standard form is $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
2. The parameter $a$ in the equation determines the orientation of the parabola: if $a > 0$, the parabola opens upward; if $a < 0$, the parabola opens downward.
3. The vertex of a parabola is the point where the parabola changes direction and represents the minimum or maximum value of the function.
4. The axis of symmetry of a parabola is the vertical line that passes through the vertex, dividing the parabola into two equal halves.
5. Parabolas are often used to model real-world phenomena, such as the trajectory of a projectile, the shape of a satellite dish, or the cross-section of a suspension bridge.

Review Questions

• Explain the relationship between a parabola and a quadratic function.
  • A parabola is the graph of a quadratic function, which is a polynomial function of degree two. The equation of a parabola in standard form is $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The parameter $a$ determines the orientation of the parabola, with $a > 0$ indicating an upward-opening parabola and $a < 0$ indicating a downward-opening parabola. The vertex of the parabola represents the minimum or maximum value of the quadratic function.
• Describe the key features of a parabola, including the vertex and axis of symmetry.
  • The key features of a parabola include the vertex and the axis of symmetry. The vertex is the point on the parabola where the curve changes direction, representing the minimum or maximum value of the function. The axis of symmetry is the vertical line that passes through the vertex, dividing the parabola into two equal halves. These features are important in understanding the behavior and properties of a parabolic curve, as they provide information about the function's extrema and the overall shape of the curve.
• Analyze how parabolas are used to model real-world phenomena, and explain the significance of this application.
  • Parabolas are widely used to model various real-world phenomena due to their unique shape and mathematical properties. For example, the trajectory of a projectile, such as a ball or a rocket, can be approximated by a parabolic curve, which is determined by the initial velocity, angle of launch, and the effects of gravity. Additionally, parabolas are used to describe the shape of satellite dishes, the cross-section of suspension bridges, and the path of water in a fountain. The ability to model these phenomena using parabolas allows for the analysis, prediction, and optimization of various engineering and scientific applications, making them an important tool in various fields.

Related terms

Quadratic Function: A polynomial function of degree two, in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Vertex:

The point on a parabola where the curve changes direction, representing the minimum or maximum value of the function.

Axis of Symmetry: The vertical line that passes through the vertex of a parabola, dividing the curve into two equal halves.