Question

The graph of\({\bf{f}}\)is given. Use it to graph the following functions.

(a)\({\bf{y = f}}\left( {{\bf{2x}}} \right)\)

(b)\({\bf{y = f}}\left( {\frac{{\bf{1}}}{{\bf{2}}}{\bf{x}}} \right)\)

(c)\({\bf{y = f}}\left( {{\bf{ - x}}} \right)\)

(d)\({\bf{y = - f}}\left( {{\bf{ - x}}} \right)\)

Short answer

The graph of the given functions is given below.

Step-by-step solution

Given data

Consider the graph of a function \(f\) as shown as below:

Redraw the graph of \(f\) say with points as follows:

(a) \({\bf{y = f}}\left( {{\bf{2x}}} \right)\)

The objective is to graph the function \(y = f\left( {2x} \right)\)by using of the function \(y = f\left( {cx} \right)\)on\(f\).

Consider \(c > 1\).

To obtain the graph of\(y = f\left( {cx} \right)\)shrink the graph of\(y = f\left( x \right)\)horizontally by a factor of c.

Here,\(c = 2 > 1\).

To obtain the graph of\(y = f\left( {2x} \right)\)shrink the graph of\(y = f\left( x \right)\)horizontally by a factor of 2.

Thus, the required graph is;

:( b) \({\bf{y = f}}\left( {\frac{{\bf{1}}}{{\bf{2}}}{\bf{x}}} \right)\)

The objective is to graph the function\(y = f\left( {\frac{1}{2}x} \right).\)

Consider \(c > 1\).

To obtain the graph of\(y = f\left( {\frac{x}{c}} \right)\)shrink the graph of\(y = f\left( x \right)\)vertically by a factor of c.

Here,\(c = 2 > 1\)

To obtain the graph of\(y = f\left( {\frac{x}{2}} \right)\)shrink the graph of\(y = f\left( x \right)\)vertically by a factor of 2.

Thus, the required graph is,

(c) \({\bf{y = f}}\left( {{\bf{ - x}}} \right)\)

The objective is to graph the function \(y = f\left( x \right)\)by using the graph of the function\(f\).

To obtain the graph of\(y = f\left( { - x} \right)\)reflecting the graph of\(y = f\left( x \right)\)about the y-axis.

The required graph is as shown below;

(d) \({\bf{y =  - f}}\left( {{\bf{ - x}}} \right)\)

The objective is to graph the function by \(y = - f\left( { - x} \right)\)by using the graph of the function\(f\).

To obtain the graph of\(y = - f\left( { - x} \right)\)reflecting the graph of\(y = f\left( { - x} \right)\)about the x-axis.

The required graph is as shown below: