TRANSFORMATION OF GRAPH FROM THE GIVEN FUNCTION

Here we are going to see, how to graph the function using transformations. 

Reflection :

A reflection is the mirror image of the graph where line l is the mirror of the reflection.

Here f' is the mirror image of f with respect to l. Every point of f has a corresponding image in f'. Some useful reflections of y = f(x) are

(i) The graph y = −f(x) is the reflection of the graph of f about the x-axis.

(ii) The graph y = f(−x) is the reflection of the graph of f about the y-axis.

(iii) The graph of y = f⁻¹(x) is the reflection of the graph of f in y = x.

Translation :

A translation of a graph is a vertical or horizontal shift of the graph that produces congruent graphs.

The graph of

y = f(x + c), c > 0 causes the shift to the left.

y = f(x − c), c > 0 causes the shift to the right.

y = f(x) + d, d > 0 causes the shift to the upward.

y = f(x) − d, d > 0 causes the shift to the downward.

Consider the functions:

(i) f(x) = |x| (ii) f(x) = |x − 1| (iii) f(x) = |x + 1|

Dilation :

Dilation is also a transformation which causes the curve stretches (expands) or compresses (contracts). Multiplying a function by a positive constant vertically stretches or compresses its graph; that is, the graph moves away from x-axis or towards x-axis. 

If the positive constant is greater than one, the graph moves away from the x-axis. If the positive constant is less than one, the graph moves towards the x-axis.

Consider the functions:

(i) f(x) = x² (ii) f(x) = (1/2) x² (iii) f(x) = 2x²

Practice Question

From the curve y = x, draw

(i) y = −x

(ii) y = 2x

(iii) y = x + 1

(iv) y = (1/2)x + 1

(v) 2x + y + 3 = 0.

Solution :

(i) y = −x

Given function :

y = x

put x = -x 

y = -x

y = - f(x)

Hence the graph y = -x is the reflection of the graph y =x.

(ii) y = 2x

Here the constant 2 is multiplied with x, so we have to perform dialation.

Since the positive constant is greater than one, the graph moves away from the x-axis 2 units. 

(iii) y = x + 1

Since 1 is added to the function, we have to translate the graph of y = x 1 unit upward.

(iv) y = (1/2)x + 1

Step 1 :

Since 1/2 is multiplied by x, we have to perform translation. If the positive constant which is multiplied by x is less than one, the graph moves towards the x-axis.

Step 2 :

1 is added to f(x), we have to move the graph 1 units upward.

(v) 2x + y + 3 = 0

y = -2x - 3

First let us consider y = -2x, it is the reflection of y = 2x about x axis.

Now we have to to subtract 3 from -2x, so we have move the curve 3 units below.

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