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.

13. \(y = {\bf{1}} - \frac{{\bf{1}}}{{\bf{2}}}{e^{ - x}}\)

Short answer

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

• Reflect about the \(y\)-axis.
• Then, compress the graph vertically by factor 2.
• And, then reflect it about the \(x\)-axis.
• At last, reflect the graph 1 unit upward.

Step-by-step solution

Write the transformation to find the graph of \(y = {\bf{1}} - \frac{{\bf{1}}}{{\bf{2}}}{e^{ - x}}\)

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

• Reflect about the \(y\)-axis.
• Then, compress the graph vertically by factor 2.
• And, then reflect it about the \(x\)-axis.
• At last, reflect the graph 1 unit upward.

Sketch the graph of transformation from \(y = {e^x}\), \(y = {e^{ - x}}\), \(y = \frac{{\bf{1}}}{{\bf{2}}}{e^{ - x}}\), \(y =  - \frac{{\bf{1}}}{{\bf{2}}}{e^{ - x}}\) and \(y = {\bf{1}} - \frac{{\bf{1}}}{{\bf{2}}}{e^{ - x}}\)

The figure below represents the graph of \(y = {e^x}\):

The figure below represents the graph of \(y = {e^{ - x}}\):

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

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

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

Thus, the graph of the function is obtained.