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From the point of construction of the graph of y=|f(x)|, we need to modify the graph of y=f(x) as :

(i) take the mirror image of lower half of the graph in x-axis

(ii) remove lower half of the graph

This completes the construction for y=|f(x)|.

Problem : Draw graph of y = | cos x | .

Solution : We first draw the graph of y = cos x . Then, we take the mirror image of lower half of the graph in x-axis and remove lower half of the graph to complete the construction of graph of y = | cos x |

Modulus operator applied to cosine function

Modulus operator applied to cosine function.

Problem : Draw graph of y = | x 2 2 x 3 |

Solution : We first draw graph y = x 2 2 x 3 . The roots of corresponding quadratic equation are -1 and 3. After plotting graph of quadratic function, we take the mirror image of lower half of the graph in x-axis and remove lower half of the graph to complete the construction of graph of y = | x 2 2 x 3 |

Modulus operator applied to quadratic function

Modulus operator applied to quadratic function.

Problem : Draw graph of y = | log 10 x | .

Solution : We first draw graph y = log 10 x . Then, we take the mirror image of lower half of the graph in x-axis and remove lower half of the graph to complete the construction of graph of y = | log 10 x | .

Modulus operator applied to logarithmic function

Modulus operator applied to logarithmic function.

Modulus function applied to dependent variable

The form of transformation is depicted as :

y = f x | y | = f x

As discussed in the beginning of module, value of function is first calculated for a given value of x. The value so evaluated is assigned to the modulus function |y|. We interpret assignment to |y| in accordance with the interpretation of equality of the modulus function to a value. In this case, we know that :

| y | = f x ; f x > 0 y = ± f x

| y | = f x ; f x = 0 y = 0

| y | = f x ; f x < 0 Modulus can not be equated to negative value. No solution

Clearly, we need to neglect all negative values of f(x). For every positive value of f(x), there are two values of dependent expressions -f(x) and f(x). It means that we need to take image of upper part of the graph across x-axis. This is image in x-axis.

From the point of construction of the graph of |y|=f(x), we need to modify the graph of y=f(x) as :

1 : remove lower half of the graph

2 : take the mirror image of upper half of the graph in x-axis

This completes the construction for |y|=f(x).

Problem : Draw graph of | y | = x 1 x 3 .

Solution : We first draw the graph of quadratic function given by y = x 1 x 3 . Then, we remove lower half of the graph and take mirror image of upper half of the graph in x-axis to complete the construction of graph of | y | = x 1 x 3 .

Modulus operator applied to dependent variable

Modulus operator applied to dependent variable.

Problem : Draw graph of | y | = tan - 1 x .

Solution : We first draw the graph of function given by y = tan - 1 x . Then, we remove lower half of the graph and take mirror image of upper half of the graph in x-axis to complete the construction of graph of y = tan - 1 x .

Modulus operator applied to dependent variable

Modulus operator applied to dependent variable.

Modulus function applied to inverse function

The form of transformation is depicted as :

Questions & Answers

What is power set
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Period of sin^6 3x+ cos^6 3x
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Period of sin^6 3x+ cos^6 3x
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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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