In plane geometry, a parabola is a curve composed of all points on the plane with equal distance from a fixed point (focus). A parabola also includes a straight line (called a directrix), which is perpendicular to the connecting line from a fixed point to every point on the curve.
A parabola is divided into two symmetrical parts by an axis of symmetry, which is a straight line perpendicular to the directrix and passing through the focal point. The shape of parabola can be determined according to the opening direction of parabola and the distance from focus to vertex. If the opening of the parabola is upward, the focus is above the vertex of the parabola; If the opening of the parabola is downward, the focus is below the vertex of the parabola.
Parabola can be expressed in the standard form of quadratic function as y = ax? +bx+c, where a, b and c are constants and a is not equal to zero. The vertex coordinates of parabola can be obtained by solving the vertices of quadratic function.
Parabola is widely used in mathematics and physics, such as describing the trajectory of free fall in mechanics and reflection and refraction in optics.
Application of parabola
As a common curve shape, parabola is widely used in many fields. Here are some examples of parabolas:
1. Physics and Engineering: Parabola is widely used to describe the trajectory of free fall. For example, the thrown object moves along a parabolic path under the action of gravity. This has important applications in throwing, shooting, projectile sports and other fields. In addition, the parabolic antenna also uses the parabolic shape to focus electromagnetic waves on one point.
2. Architectural design and urban planning: Parabola is often used in the design of buildings and bridges. For example, the sections in domes, arches and colonnades usually adopt parabolic shapes to provide good structural stability and uniform force transmission. In addition, fountains and fountains in garden design often use parabolic shapes to achieve aesthetic effects.
3. Projection and photography: In projectors and cameras, parabolic mirrors are used to focus light on one point to obtain clearer and brighter images. This focusing mechanism is often used in telescopes, astronomical observation instruments and lasers.
4. Projectile and trajectory: Parabola is also used to predict the trajectory of objects. For example, in projectiles such as shells, rockets or golf balls, the mathematical model of parabola can be used to calculate the flight trajectory and landing point of projectiles.
5. Mathematical modeling and computer graphics: Parabola also has important applications in mathematical modeling and computer graphics. By using parabolic equation and parametric curve, we can realize the tasks of modeling complex shapes, generating animation effects and image processing.
These are just some examples of parabolic applications. In fact, parabola is widely used in science, engineering, art and daily life.
Examples of parabolas
Example 1:
Given the parabolic equation y = 2x? -4x+ 1, find the vertex coordinates and focus coordinates of the parabola.
Answer:
First of all, we can find the vertex coordinates of parabola by solving the vertices of quadratic function. The abscissa of the vertex of a parabola can be obtained by the formula x = -b/(2a).
In this example, a = 2 and b = -4. Substituting these values into the formula, we can get x = -(-4)/(2*2) = 1.
Substituting x = 1 into the parabolic equation, we can calculate the value of y: y = 2( 1)? - 4( 1) + 1 = - 1。
So the vertex coordinates of parabola are (1,-1).
Secondly, the focal coordinates can be determined directly by the focal formula. In the general parabolic equation, the form is y = ax? In +bx+c, the abscissa of the focus can be expressed as x = -b/(2a).
For the parabolic equation y = 2x in this example? -4x+ 1, we have a = 2 and b = -4, so the abscissa of the focus is x = -(-4)/(2*2) = 1.
Substituting x = 1 into the parabolic equation, we can calculate the ordinate of the focus: y = 2( 1)? - 4( 1) + 1 = - 1。
Therefore, the focal coordinates are (1,-1).
So the vertex coordinates and focus coordinates of this parabola are (1,-1).
Example 2:
It is known that the focus of the parabola is at point F (-3,2), and the directrix of the parabola coincides with the X axis. Find the equation of parabola.
Answer:
Because the directrix coincides with the X axis, it shows that the equation of the directrix of parabola is y = 0.
Because the focus is at point F (-3,2), according to the definition of parabola, the distance from any point on the parabola to the focus is equal to the distance from the point to the sight line.
Consider the focus f (-3,2), the directrix y = 0, and let a point on the parabola be P(x, y). According to the distance formula, we can get the following equation:
√[(x-(-3))？ + (y-2)？ ] = |y - 0|
After simplification, you can get:
(x+3)？ + (y-2)？ = y？
Expand and arrange to obtain:
x？ + 6x + 9 + y？ - 4y + 4 = y？
Simplified to:
x？ + 6x + 13 = 4y
Therefore, the equation of parabola is x? + 6x + 13 = 4y .