GRAPH THE FUNCTION USING TRANSFORMATIONS EXAMPLES

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

Examples

Example 1 :

Graph the functions f(x) = x3 and g(x) = 3√x on the same coordinate plane. Find f ◦ g and graph it on the plane as well. Explain your results.

Solution :

f(x) = x3 and g(x) = 3√x

f ◦ g (x)  =  f [ g (x) ]

   =  f (x1/3)

  =   (x1/3)3

f ◦ g (x)  =  x

y =  x3

if x = -2, then y = -8

if x = -1, then y = -1

if x = 0, then y = 0

if x = 1, then y = 1

if x = 2, then y = 8

y =  x1/3

if x = -8, then y = -2

if x = -1, then y = -1

if x = 0, then y = 0

if x = 1, then y = 1

if x = 8, then y = 2

f(x) = x3 

Let y = x3 

Let us find the inverse function, for that we have to solve for x.

x = y1/3

Now we have to replace "x" by "f-1(x)" and "y" by "x".

f-1(x)  =  x1/3

By finding inverse of the given function, we get the other function.

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

Example 2 :

Write the steps to obtain the graph of the function y = 3(x − 1)2 + 5 from the graph y = x2

Solution :

Step 1 :

By graphing the curve y = x2, we get a open upward parabola with vertex (0, 0).

Step 2 :

Here 1 is subtracted from x, so we have to shift the graph of y = x2, 1 unit to the right side. 

Step 3 :

The positive number 3 is multiplied by (x-1) which is greater than 1, so we have to compress the curve y = (x-1)2 towards y-axis.

Step 4 :

5 is added to the function, so we have to move the graph of  y = 3(x-1)2, 5 units to the left side.

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