Question

The graph of\({\bf{f}}\)is given. Draw the graphs of the following functions.

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

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

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

(d)\({\bf{y = f}}\left( {\frac{{\bf{1}}}{{\bf{3}}}{\bf{x}}} \right){\bf{ + 1}}\)

Short answer

The graph of each function is given below.

Step-by-step solution

Given data

We are given the basic graph of \(y = f\left( x \right)\) red below. In all the graphs the basic graph will always stay the color red. The transformed graphs will have different colors.

:(a) The transformed graph is given by\({\bf{y = f}}\left( {\bf{x}} \right){\bf{ - 2}}\)\(\)

This shifts the basic graph by 2 units downward. So take every point above in red and shift it down by 2 to get the blue graph below.

(b) The transformed graph is given by \({\bf{y = f}}\left( {{\bf{x - 2}}} \right)\)

This shifts the basic graph by 2 units to the right. So take every point above in red and shift it to the right by 2 to get the blue graph below.

(c) The transformed graph is given by\({\bf{y =  - 2f}}\left( {\bf{x}} \right)\)\(\)

This has two transformations on the basic red graph. First, reflect the basic graph about the \(x - {\rm{axis}}\) (seen in green in green below), then stretch the graph vertically by 2 units. The final graph is in blue.

Note: The act of stretching vertically by a factor of 2 means taking each \(y - coordinate\)in green and multiplying by 2.

(d) The transformed graph is given by\({\bf{y = f}}\left( {\frac{{\bf{1}}}{{\bf{3}}}{\bf{x}}} \right){\bf{ + 1}}\)\(\)

This has two transformations on the basic red graph. Firstly, stretch the basic red graph horizontally by\(\frac{1}{3}\), and then shifts the graph upward by 1 unit.

Note: The act of stretching horizontally by a factor of 3 means taking each\(x - {\rm{coordinate}}\)in red and multiplying by 3.\(\)