Vertical and Horizontal Transformation: Definition & Equation

Understanding how the function transforms is important in mathematics. Functional transformation alters the position and size of the graph of the function. Transformation can be of several types like Horizontal transformation and vertical transformation. It involves moving graphs up and down and helps the analyst to find different insights.

Functions:

A relation can be defined as a relationship between two or more set of information.A relation from a set $A$ to a set $B$ is a function from $A$ to $B$ if every element of set $A$ has one and only one image in set $B$.

OR

$A$ and $B$ are two non-empty sets, then a relation from $ A$ to $B$ is said to be a function if each element $x$ in $A$ is assigned a unique element $f(x)$ in $B$, and it is written as

$f: A ➝ B$ and read as $f$ is mapping from $A$ to $B.$

Vertical and Horizontal Transformation

Vertical shift $f(x) → f(x) ± a$

A vertical shift of a function occurs if we add or subtract the constant to the function $y = f(x)$

For $a > 0$, the graph of $y = f(x) + a$ is obtained by shifting the graph of $f(x)$ upwards by $‘a’$ units,

whereas the graph of $y = f(x) − a$ is obtained by shifting the graph of $f(x)$ downwards by ‘$a$’ units.

For Example:

The graph of the function $f(x)=x^2+4$ is the graph off $(x)=x^{\wedge} 2$ shifted up by 4 units;

The graph of the function $f(x)=x^2-4$ is the graph of $f(x)=x^2$ shifted down by $4$ units.

Horizontal shift: $f(x) → f(x±a)$

A horizontal shift of a function occurs if we add or subtract the same constant to each input $x$.

For $a > 0$, the graph of $y = f(x + a)$ is obtained by shifting the graph of $f(x)$ to the left by ‘a’ units.

The graph of $y = f(x − a)$ is obtained by shifting the graph of $f(x)$ to the right by ‘$a$’ units.

For Example

$f(x) = |x + 3|$

The graph of $f(x) = |x + 3|$ is the graph of $y = |x|$ shifted leftwards by $3$ units. Similarly, the graph of $f(x) = |x − 3|$ is the graph of $y = |x|$ shifted rightward by $3$ units

Solved Examples Based on Vertical and Horizontal Transformation:

Example 1: The area bounded by the lines $y=|| x-1|-2|$ and $\mathrm{y}=2$ is

1) $8$

2) $10$

3) $12$

4) $6$

Solution

Given the equation of curve are

$y = ||x-1|-2|$

and, $y = 2$

Plot the curve on the graph

We have to find area of triangle $ACD$ and triangle $BDE$

$\begin{aligned} & \text { Area }=\frac{1}{2} \times 2 \times C D+\frac{1}{2} \times 2 \times D E \\ & \text { Area }=C D+D E=8\end{aligned}$

Example 2: Which of the following is the graph of $y = |x| + 5$?

1)

2)

3)

4)

Solution

As we have learnt in

Vertical and Horizontal Transformation -

For $a > 0$, the graph of $y = f(x) + a$ is obtained by shifting the graph of $y = f(x)$ upwards by ‘$a$’ units

The graph of $y=|x|$

Now to draw the graph of $y=|x|+5$, the graph of $|x|$ is shifted upwards by $5$ units as shown below

Summary

Understanding these transformations can help solve real-life complex problems into simpler ones like signal processing, demand and supply curves etc. This operation is associative and plays an important role in various mathematical and applied fields, including calculus, function transformation, and computer science.