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**Parabola|Definition & Meaning**

**Definition**

A **plane** and a **cone’s** **junction** **result** in a **curve** known as a **parabola**. It is a **particular** **kind** of **conic section,** which are curves made when a plane and a cone cross. The **focus** (a point on the curve) and **directrix determine** the **parabola’s shape** (a line). The **collection** of all **points** that are **equally spaced apart** from the **directrix** and **focus** is **known** as a **parabola.**

**Components of Parabola **

Figure 1 – Components of Parabola

The components of the parabola are illustrated in the figure above.

**Axis of symmetry:**The**line**that**splits**the**parabola into**two**identical halves**is known as the**axis**of**symmetry.**The**directrix**and the axis of**symmetry**are**always perpendicular.****Vertex:**The**parabola’s highest**or**lowest point**is known as the**vertex.**The parabola’s vertex is the point closest to the focus.**Focus:**The**location on**the axis of**symmetry**that**serves**as the**parabola’s definition.**All the points that are equally spaced apart from the focus and the directrix make up the parabola.**Directrix:**The**line**that**serves**as the parabola’s definition. All the points that are equally spaced apart from the focus and the directrix make up the parabola.**Latus Rectum:**The**latus rectum**is the**portion**of the**parabola**that**passes through**the**focus**while being**perpendicular to**the axis of**symmetry.**The latus rectum**determines**the**size**and**shape**of the parabola.**Conjugate axis:**The**axis**that**traverses**the**focus**while running perpendicular to the axis of symmetry. The parabola’s size and shape are determined by the conjugate axis.

**Different Equations for Parabola**

There are several different forms of the equation of a parabola that can be used, depending on the specific needs of the problem at hand.

**Standard form:**y = a(x – l)$^{2}$ + m, where (l, m) denotes the vertex of the parabola.**In Vertex form:**y = a(x – l)$^{2}$, where (l, k) denotes the vertex of the parabola. This form is useful when the focus of the parabola is known and the vertex form is needed.**General form:**y = ax$^{2}$ + bx + c, where**a, b, and c**are**constants.**This form is**useful when**the line of**symmetry,**or axis of symmetry, of the parabola, is**known.****Factorized form:**y = a(x – b)(x – c), where a, b, and c are constants and r and s are the roots of the equation. This form is**useful when**the**x-intercepts**of the parabola are**known.****Parametric form:**x = at$^{2}$ and y = 2at + b, where a, b, and t are constants. This form is**useful when**the**focus**of the parabola is**known**and the vertex form is not needed.**Polar form:**r = at$^{2}$, where a and t are constants. This form is useful when the**parabola**is in**polar coordinates.**

**Types of Parabola**

There are **two main types** of parabolas: those that **open upwards** and those that **open downwards.** These types are illustrated in the figure shown below.

1. A **mathematical equation** of the form y = ax$^{2}$ + bx + c, where a > 0, **defines** an **upward-opening** parabola. This equation’s graph will be a parabola with an upward opening and a vertex at the bottom of the curve.

Figure 2 – Parabola Opening Upward

2. An **equation** of the **form** y = ax$^{2}$ + bx + c, where a < 0, describes a **downward-opening parabola.** The graph of this equation will be a downward-opening parabola, with the highest point on the curve serving as the vertex.

Figure 3 – Parabola Opening Downward

**Graph of Parabola and Properties**

The graph of a parabola has several important features.

Figure 4 – Graph of Parabola

- The
**vertex**is**either**the**highest**point on the curve**or**its**lowest**point, depending on how the parabola opens. It is situated at the coordinates (h, k), where h and k represent the x- and y-values, respectively. - The
**axis**of**symmetry**is the line that**divides**the**parabola**into**two halves**that are mirror images of one another. It runs across the vertex and is a line of symmetry for the parabola. - While the
**focus**is a**fixed point inside**the**parabola,**the**directrix**is a**line**that is**constantly outside**the**parabola.**The**focus**and the**directrix**are**equally distant**from each point on the parabola. **Depending**on the**values**of the**variables**a, b, and c in the equation, the**parabola**may**intersect**either the**x-axis**or the**y-axis.**

**Application of Parabola**

**Parabolas** have a **wide** range of **applications** in various fields, including mathematics, science, engineering, and technology. Here are a few examples:

**Mathematicians**use parabolas to**represent**and**examine**a wide range of**phenomena, including**the**trajectory**of a**bullet**and the**power**of**sound waves.**- In
**physics,**parabolas are used to explain**how objects move**when they are**affected by gravity.**A parabolic curve, for instance, describes the**path**of a**ball**being thrown or a satellite in orbit. - When
**designing structures**and equipment for the**precise focus**or reflection of light, sound, or other types of energy,**engineers use parabolas.**Headlamps, parabolic microphones, and satellite dish antennae are a few examples. **Buildings**that can**survive earthquakes**and other types of structural stress are**designed using parabolas**in architecture.**Parabolas**are used in**finance**to**simulate, examine,**and make**investment decisions**based on the behavior of the**financial markets.**- In
**sports,**such as**golf, soccer,**and**baseball, parabolas**are**used**to**study**and**optimize**the**trajectory**of**balls**and other items.

**Parabola in Conic Section**

A **parabola** is a **particular kind** of **conic section,** a curve created by the intersection of a right circular cone and a plane. The **ellipse, hyperbola,** and **circle** are the **other shapes** that can have conic sections.

A parabola is the **collection** of all **points** that are **equidistant** from a particular point (the **focus)** and a particular line (the **directrix).** It can also be **envisioned** as the **perpendicular** to the **cone’s axis cross-section** of a right circular cone.

**Solved Example of a Parabola**

**Example**

**Locate** the **parabola’s vertex** according to the **equation given** by y = x$^{2}$ + 2x – 3.

**Solution**

To **find** the **vertex** of the parabola, we can **rewrite** the **equation** in vertex form: y = a(x – h)$^{2}$ + k.

To do this, we can **complete** the **square:**

y = x$^{2}$ + 2x – 3 = (x$^{2}$ + 2x) – 3 = x$^{2}$ + 2x + 1 – 4 = (x + 1)$^{2}$ – 3

So, the **vertex** of the **parabola** is at the **point (h,****k) = (-1, -3).**

*All mathematical drawings and images were created with GeoGebra.*