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Translations of a Graph

We study about graphs in geometry. A graph is said to be a mathematical structure exhibiting the relationships between two or more variables or two or more sets of values or measurements.

Geometry is a subject that is interconnected with almost all the branches of mathematics. Calculus and geometry are two interrelated branches of mathematics. In calculus, the translation of geometrical graphs is an important chapter.

In mathematical language, translation means moving a graph either horizontally or vertically or both. In other words, when each point on a graph is shifted towards left or right or up or down, this process is known as translation.

It is a very important concept which have been utilized widely in middle and higher-level mathematics. The function of the graph which is translated becomes completely a different one after translation. In this article, let us go ahead and learn about translation of graphs in detail.

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Definition

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The translation is defined as shifting of the graph. We can say that the shift of each point of a graph by some constant number either horizontally or vertically, is known as translation of that graph. Translation is the result of either addition operation or subtraction operation to the function. 

In this way, there are mainly two types of translations - horizontal and vertical. By just looking at the equation of the graph, we can find out whether it is a horizontal translation or vertical translation.

Horizontal Translation:
Horizontal translation of a graph is defined as the shifting of each point on the graph towards either left or right side. Thus, shifting on graph towards the direction of X axis, is called horizontal translation.
If the base (original) graph is given by the equation y = f(x), then the equation representing its translation k units horizontally is given below :

y$_{1}$ = f(x - k)

Here, if k > 0, then the base graph is to be translated k units towards the right side, i.e. towards positive X axis.
if k < 0, then the base graph is to be translated k units towards the left side, i.e. towards negative X axis.


For Example:

1) Given that y = |x|

then y = |x - 3| indicates the horizontal transformation towards right as shown in the following graph.

Translations of a Graph

Here, blue line represents the base graph y = |x |
while red line denotes the transformed graph y = |x - 3|.

2)  Given that y = 5 $\sqrt{x}$
then y = 5 $\sqrt{x + 2}$ indicates the horizontal transformation towards left as shown in the graph below.
Graph of Translations

Here, green line shows the base graph y = 5 $\sqrt{x}$
while blue line shows the transformed graph y = 5 $\sqrt{x + 2}$.

Vertical Translation:
The shifting of every point on the graph upwards or downwards is defined as the vertical translation. In other words, when the whole is shifted in the direction of Y axis, either up or down, it is known as vertical translation.
If the equation of base graph is y = f(x), then the equation which represents its translation k units vertically, is:
y$_{1}$ = f(x) + k

Note that if k > 0, then the base graph should be translated k units in upward direction, i.e. towards positive Y axis.
Also, if k < 0, then the base graph should be translated k units in downward direction, i.e. towards negative Y axis.
For Example:
1) Given function is y = 4x

then y = 4x + 1 indicates the vertical transformation upwards. It is shown in the following graph.
Vertical Translations

Here, red line represents the base graph y = 4x
while orange line denotes the transformed graph y = 4x + 1.

2)  Given that y = x

then the equation y = x - 3 would be the vertical transformation downwards as shown in the figure below.
Vertical Translations of a Graph

Here, blue line shows the base graph y = x
and green line denotes the transformed graph y = x - 3.

Translation of Center of Circle

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We know that a circle is a plane figure, on which all the points are at same distance from one fixed point, known as center of the circle.

Also, this same distance is called radius of the circle.
 
The equation of a circle is given below:

$x^{2} + y^{2} = r^{2}$

This equation indicates that it is circle with center located at (0, 0) and radius equal to r.

The center of the circle can be translated from origin to some other point. The equation with center at (h, k) will be denoted by:

$(x - h)^{2} +  (y - k)^{2} = r^{2}$

Where, h and k are x coordinate and y coordinate of the center.
For Example:
The circle $(x - 3)^{2} +  (y - 4)^{2} = 1$ is a circle of radius 1 unit and center located at point (3, 4) as shown below:

Translations of Center of Circle

Translation of Vertex of Parabola

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The equations of the parabola are given by

y$^{2}$ = 4ax

x$^{2}$ = 4ay

Where, the vertex of the parabola is located at origin (0, 0).

If the equations of the parabola be:

(y - k)$^{2}$ = 4a (x - h)

(x - h)$^{2}$ = 4a (y - k)

It means that the origin of the parabola is shifted from origin to (h, k).

For Example:

A parabola y$^{2}$ = 2x is shown below.
Translations of Vertex of Parabola

Also, the parabola  (y+1)$^2$  = 2 (x + 3) has its vertex at (-3, -1). Have a look.

Translations of Vertex of Parabola Example

Vertical Stretches and Shrinks

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Stretching of a graph basically means pulling the graph outwards. Also, by shrinking a graph, we mean compressing the graph inwards. Stretching and shrinking changes the dimensions of the base graph, but its shape is not altered. While vertical stretch and shrink, the x-intercepts remain unchanged.
If the base function be f (x) and there is a constant number such that k > 0, then the function which gives vertical stretch and shrink of f(x) is as follows:

g(x) = k . f(x)

Here, if k > 1, then the graph will be k units vertically stretched.

And, if k < 1, then the graph will be k units vertically shrunk. (note that k is a nonzero number)
For Example: Take a look at the graphs graph f(x) = $x^2$ - 2 and g(x) = $\frac{1}{2}$
 ($x^2$
- 2).
Vertical Stretches and Shrinks

The graph 
g(x) = $\frac{1}{2}$ ($x^2$ - 2) shirks vertically by a factor of $\frac{1}{2}$.
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