Question

The graph of\({\bf{x = f}}\left( {\bf{x}} \right)\)is given. Match each equation with its graph and give reasons for your choices.

(a)\({\bf{y = f}}\left( {{\bf{x - 4}}} \right)\)

(b)\({\bf{y = f}}\left( {\bf{x}} \right){\bf{ + 3}}\)

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

(d)\({\bf{y = - f}}\left( {{\bf{x + 4}}} \right)\)

(e)\({\bf{y = 2f}}\left( {{\bf{x + 6}}} \right)\)

Short answer

(a) Graph 3 represents the equation\({\bf{y = f}}\left( {{\bf{x - 4}}} \right)\)

(b)Graph 1 represents the function\(y = f\left( x \right) + 3\).

(c) Graph 4 represents the function\(y = \frac{1}{3}f\left( x \right)\).

(d) Graph 5 represents the function\(y = - f\left( {x + 4} \right)\).

(e) Graph 2 represents the function\(y = 2f\left( {x + 6} \right)\).

Step-by-step solution

Given data

Consider the graph,

(a) \({\bf{y = f}}\left( {{\bf{x - 4}}} \right)\)

Consider the function, \(y = f\left( {x - 4} \right)\)

To obtain the graph of\(y = f\left( {x - 4} \right)\), shift the graph\(y = f\left( x \right)\)distance 4 units to the right.

From the figure, observe that graph 3 represents the function\(y = f\left( {x - 4} \right)\).

(b) \({\bf{y = f}}\left( {\bf{x}} \right){\bf{ + 3}}\)

Consider the function,\(y = f\left( x \right) + 3\)\(x\)

To obtain the graph of \(y = f\left( x \right) + 3\), shift the graph\(y = f\left( x \right)\)upwards by 3 units.

From the figure, observe that graph 1 represents the function\(y = f\left( x \right) + 3\).

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

Consider the function,\(y = \frac{1}{3}f\left( x \right)\)\(\)\(y = f\left( x \right) + 3\)

To obtain the graph of\(y = \frac{1}{3}f\left( x \right)\), shrink the graph\(y = f\left( x \right)\)vertically by a factor of\(\frac{1}{3}\).

From the figure, observe that graph 4 represents the function\(y = \frac{1}{3}f\left( x \right)\).

(d) \({\bf{y =  - f}}\left( {{\bf{x + 4}}} \right)\)

Consider the function, \(y = - f\left( {x + 4} \right)\)\(y = - f\left( {x + 4} \right)\).

To obtain the graph of\(y = - f\left( {x + 4} \right)\), reflect the graph\(y = f\left( x \right)\)about the\(x\)-axis and shift the graph\(y = f\left( x \right)\)by 4 units to the left.

From the figure, observe that graph 5 represents the function\(y = - f\left( {x + 4} \right)\).

(e) \({\bf{y = 2f}}\left( {{\bf{x + 6}}} \right)\)\(\)

Consider the function,\(y = 2f\left( {x + 6} \right)\).

To obtain the graph of\(y = 2f\left( {x + 6} \right)\), shift the graph\(y = f\left( x \right)\)by 6 units to the leftand stretch the graph of\(y = f\left( x \right)\)vertically by a factor of 2.

From the figure, observe that graph 2 represents the function\(y = 2f\left( {x + 6} \right)\).

Conclusion of the graph