Question

Make a rough sketch by hand of the graph of the function. Use the graphs given in Figures 3 and 15 and, if necessary, the transformations of Section 1.3.

10. \(h\left( x \right) = {\bf{2}}{\left( {\frac{{\bf{1}}}{{\bf{2}}}} \right)^x} - {\bf{3}}\)

Short answer

The graph of the function \(h\left( x \right) = 2{\left( {\frac{1}{2}} \right)^x} - 3\) is obtained from the graph of \(h\left( x \right) = {\left( {\frac{1}{2}} \right)^x}\) after the following transformation:

• Stretch the graph by a factor of 2.
• Then, shift the graph 3 units downward.

Step-by-step solution

Write the transformation to find the graph of \(h\left( x \right) = {\bf{2}}{\left( {\frac{{\bf{1}}}{{\bf{2}}}} \right)^x} - {\bf{3}}\)

The graph of the function \(h\left( x \right) = 2{\left( {\frac{1}{2}} \right)^x} - 3\) is obtained from the graph of \(h\left( x \right) = {\left( {\frac{1}{2}} \right)^x}\) after the following transformation:

• Stretch the graph by a factor of 2.
• Then, shift the graph 3 units downward.

Sketch the graph of transformation from \(h\left( x \right) = {\left( {\frac{{\bf{1}}}{{\bf{2}}}} \right)^x}\), \(h\left( x \right) = {\bf{2}}{\left( {\frac{{\bf{1}}}{{\bf{2}}}} \right)^x}\) and \(h\left( x \right) = {\bf{2}}{\left( {\frac{{\bf{1}}}{{\bf{2}}}} \right)^x} - {\bf{3}}\)

The figure below represents the graph of \(h\left( x \right) = {\left( {\frac{1}{2}} \right)^x}\):

The figure below represents the graph of \(h\left( x \right) = 2{\left( {\frac{1}{2}} \right)^x}\):

The figure below represents the graph of \(h\left( x \right) = 2{\left( {\frac{1}{2}} \right)^x} - 2\):

Thus, the graph of the function is obtained.