Parabol

A parabola is a geometric shape formed by the graph of a quadratic function, characterized by its symmetric and curved structure. This curve appears frequently in nature and engineering, playing a crucial role in fields such as physics, architecture, and optics. Key features of a parabola include its vertex, focus, and axis of symmetry, which determine its behavior and applications in various mathematical and scientific contexts.

Parabol

Let a, b, c ∈ R and a ≠ 0 . The function f : R → R is defined as follows:

f(x) = ax² + bx + c

This type of function is called a quadratic function. The graph of a quadratic function is a parabola.

General Properties of Parabolas

The graph of the quadratic function is shaped as a parabola and has the following properties:

  • If a > 0, the arms of the parabola open upwards.It takes its smallest value at the point x = 0
  • If a < 0, the arms of the parabola open downwards.It takes its maximum value at the point x = 0

The point where the function transitions from increasing to decreasing, or from decreasing to increasing, is called the vertex of the parabola. This point is denoted as T.

Vertex of a Parabola

A quadratic function’s graph has a symmetric center known as the vertex. This point represents where the function reaches its minimum or maximum value. The coordinates of the vertex are denoted as T(r, k).

Graph of Parabola for y = ax² Function

  • As the absolute value of the coefficient of x² increases, the arms of the parabola become closer to each other.
  • As the absolute value of the coefficient of x² decreases, the arms of the parabola move farther apart.

Parabola Graph and Vertex Calculation

The function f(x) = y = a(x – r)² + k represents a quadratic function, and the vertex of this parabola is determined by the point T(r, k). Let’s break down how to interpret and plot this function.

Explanation

This function has the general form of a parabola where:

  • r represents the x-coordinate of the vertex.
  • k represents the y-coordinate of the vertex.
  • a determines the direction of the parabola’s arms: if a > 0, the parabola opens upwards; if a < 0, it opens downwards.

Example Calculation

Let’s consider the quadratic function: y = f(x) = (x – 1)² + 2

  • The vertex is at T(1, 2). This means the lowest point of the parabola is at x = 1 and y = 2.
  • Since a = 1, which is greater than 0, the parabola opens upwards (the arms of the parabola point upwards).

For x = 0:

To calculate the value of y when x = 0, substitute x = 0 into the equation:

y = (-1)² + 2 = 1 + 2 = 3

This means when x = 0, the value of y is 3.

Graphing a Quadratic Function

A function given as f(x)=ax2+bx+c can be graphed by finding the following points:

1. Y-Intercept

x = 0

f(0) = y = c

Thus, the y-intercept is the point (0,c).

2. X-Intercepts

To find the x-intercepts, set y= 0 and solve the equation:

ax2+bx+c = 0

The solutions are the points where the function intersects the x-axis.

3. Vertex

The vertex of the parabola can be found using the following formulas:

Example: f(x) = x^2 - 2x - 3

1. Y-Intercept

To find the y-intercept, substitute x = 0 into the function:

f(0) = (0)2− 2(0) −3 = −3

Thus, the y-intercept is : (0,-3)

2. X-Intercepts

To find the x-intercepts, set f(x) = 0 and solve the equation:

x2−2x −3 = 0

Factoring the quadratic equation:

(x−3)(x+1) = 0

The solutions are:

x1​=3 , x2​=−1

Thus, the x-intercepts are (3,0) and (-1,0).

3. Vertex

The x-coordinate of the vertex is:

To find the y-coordinate, substitute r = 1 into the function:

f(1) = (1)2−2(1)−3 = 1−2−3 = −4

Thus, the vertex is (1,-4)

Summary

  • Y-Intercept: (0,-3)
  • X-Intercepts: (3,0) and (-1,0)
  • Vertex: (1,-4)
1 Feb, 2025