**Exploring properties of parent functions :**

In mathematics, every function can be classified as a member of a family. Each member of a family of functions is related to the simplest, or most basic, function sharing the same characteristics.This function is called the parent function.

Exploring the properties of parent functions is trying to know something about it like its graph, name, its family members etc.,

Now, let us look at some different parent functions, their family name, family members and examples.

**Family : Linear functions**

Parent function : f(x) = x

Family members : f(x) = mx + b

Examples : f(x) = 3x + 2, f(x) = -3x + 4

Special features :

(i) Straight line goes through the origin

(ii) Slope is 1

(iii) Divides the plane exactly in half diagonally

(iv) Graph only quadrants 1 and 3

Graph of the parent function f(x) = x :

**Family : Quadratic functions**

Parent function : f(x) = x²

Family members : f(x) = a(x - h)² + k

Examples : f(x) = 5(x - 3)² - 2 , f(x) = -x² + 3

Special features :

(i) Parabola that opens up ward

(ii) Vertex at the origin

(iii) y has a minimum value

(iv) y-axis is axis of symmetry

(v) Graph only in quadrants 1 and 2

Graph of the parent function f(x) = x² :

**Family : Square root functions**

Parent function : f(x) = √x

Family members : f(x) = a √(x - h) + k

Examples : f(x) = 5 √(x - 3) - 2, f(x) = - √x + 3

Special features :

(i) Semi parabola that opens right ward

(ii) Vertex at the origin

(iii) x has a minimum value

(iv) x-axis is axis of symmetry

(v) Graph only in quadrant 1

Graph of the parent function f(x) = √x :

**Family : Reciprocal function or Rational function**

Parent function : f(x) = 1/x

Examples : f(x) = 1/(x - 3), f(x) = 2/(3x + 5)

Special features :

(i) This curve has two parts

(ii) Horizontal asymptote is y = 0

(iii) Vertical asymptote is x = 0

(v) Graph only quadrants 1 and 3

Graph of the parent function f(x) = 1/x :

**Family : Absolute value function**

Parent function : f(x) = |x|

Examples : f(x) = 2|x+3|, f(x) = |x| + 5

Special features :

(i) V-shaped curve and open upward

(ii) Vertex at the origin

(iii) y has a minimum value

(iv) y-axis is axis of symmetry

(v) Graph only quadrants 1 and 2

Graph of the parent function f(x) = |x| :

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