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 (x^1/3)

  =   (x^1/3)³

f ◦ g (x)  =  x

f(x) = x³ 

Let y = x³ 

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

x = y^1/3

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

f^-1(x)  =  x^1/3

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.

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.