Question

Explain how each graph is obtained from the graph of\({\bf{y = f}}\left( {\bf{x}} \right)\).

(a)\({\bf{y = f}}\left( {\bf{x}} \right){\bf{ + 8}}\)

(b)\({\bf{y = f}}\left( {{\bf{x + 8}}} \right)\)\(\begin{array}{l}{\bf{y = f}}\left( {{\bf{x + 8}}} \right)\\{\bf{y = 8f}}\left( {\bf{x}} \right)\\{\bf{y = f}}\left( {{\bf{8x}}} \right)\\{\bf{y = - f}}\left( {\bf{x}} \right){\bf{ - 1}}\\{\bf{y = 8f}}\left( {\frac{{\bf{1}}}{{\bf{8}}}{\bf{x}}} \right)\\\end{array}\)

(c)\({\bf{y = 8f}}\left( {\bf{x}} \right)\)

(d)\({\bf{y = f}}\left( {{\bf{8x}}} \right)\)\(\)

(e)\({\bf{y = - f}}\left( {\bf{x}} \right){\bf{ - 1}}\)

(f)\({\bf{y = 8f}}\left( {\frac{{\bf{1}}}{{\bf{8}}}{\bf{x}}} \right)\)

Short answer

(a)The graph\(y = f\left( x \right) + 8\)can be obtained by shifting the basic graph\(y = f\left( x \right)\)by 8 units upward.

(b) The graph\(y = f\left( {x + 8} \right)\)can be obtained by shifting the basic graph\(y = f\left( x \right)\)by 8 units to the left.

(c)The graph of\(y = 8f\left( x \right)\)can be obtained by stretching the basic graph\(y = f\left( x \right)\)vertically by a factor of 8.

(d) The graph\(y = f\left( {8x} \right)\)shrinks the basic graph\(y = f\left( x \right)\)horizontally by a factor of 8.

(e)The graph\(y = - f\left( x \right) - 1\)can be obtained by reflecting the basic graph\(y = f\left( x \right)\)about the\(x\)-axis and shifting downward by 1 unit.

(f) The graph of \(y = 8f\left( {\frac{1}{8}x} \right)\)can be obtained by reflecting the basic graph \(y = f\left( x \right)\)horizontally and vertically by a factor of 8.

Step-by-step solution

Given data

We are given the basic graph \(y = f\left( x \right)\) in each case below.

:( a) The transformed graph is given by \({\bf{y = f}}\left( {\bf{x}} \right){\bf{ + 8}}\)

Fact: Assuming\(c > 0,y = f\left( x \right) + c\)shifts in the basic graph\(y = f\left( x \right)\)

by c units upward.

Conclusion: Thus, the graph \(y = f\left( x \right) + 8\) can be obtained by shifting the basic graph \(y = f\left( x \right)\)by 8 units upward.

(b) The transformed graph is given by\({\bf{y = f}}\left( {{\bf{x + 8}}} \right)\)

Fact: Assuming\(c > 0,y = f\left( {x + c} \right)\)shifts the basic graph\(y = f\left( x \right)\)by c units to the left.

Conclusion: Thus, the graph \(y = f\left( {x + 8} \right)\)can be obtained by shifting the basic graph \(y = f\left( x \right)\)by 8 units to the left.

(c) The transformed graph is given by\({\bf{y = 8f}}\left( {\bf{x}} \right)\)

Fact: Assuming\(c > 0,y = f\left( {x + 8} \right)\)\(c > 1,y = cf\left( x \right)\)stretches the basic graph\(y = f\left( x \right)\)

vertically by a factor of c.

Conclusion: The graph of \(y = 8f\left( x \right)\)can be obtained by stretching the basic graph \(y = f\left( x \right)\) vertically by a factor of 8.

(d) The transformed graph is given by\({\bf{y = f}}\left( {{\bf{8x}}} \right)\)\(\)

Fact: Assuming\(c > 0,y = f\left( {x + 8} \right)\)\(c > 1,y = f\left( {cx} \right)\)shrinks the basic graph\(y = f\left( x \right)\)

horizontally by a factor of c.

Conclusion: The graph of\(y = f\left( {8x} \right)\) shrinks the basic graph \(y = f\left( x \right)\)horizontally by a factor of 8.

(e) The transformed graph is given by\({\bf{y =  - f}}\left( {\bf{x}} \right){\bf{ - 1}}\)

We have two transformations occurring.

Fact:\(y = - f\left( x \right)\)reflects the basic graph\(y = f\left( x \right)\)about the\(x\)-axis.

Fact: Assuming\(c > 0,y = f\left( x \right) - c\)shifts\(y = f\left( x \right)\)by c units downward.

Conclusion: The graph\(y = - f\left( x \right) - 1\)can be obtained by reflecting the basic graph\(y = f\left( x \right)\)about the\(x\)-axis and shifting downward by 1 unit.

\(\)

(f) The transformed graph is given by\({\bf{y = 8f}}\left( {\frac{{\bf{1}}}{{\bf{8}}}{\bf{x}}} \right)\)

We have two transformations occurring.

Fact: Assuming \(c > 1,y = f\left( {\frac{1}{c}x} \right)\)stretches the basic graph \(y = f\left( x \right)\)horizontally by a factor of c.

Fact: Assuming\(y = cf\left( x \right)\)stretches the basic graph\(y = f\left( x \right)\)vertically by a factor of c.

Conclusion: The graph of \(y = 8f\left( {\frac{1}{8}x} \right)\)can be obtained by reflecting the basic graph \(y = f\left( x \right)\)horizontally and vertically by a factor of 8.