Question

Sketch the graph of the function \(y = 1 - \frac{1}{2}{e^{ - x}}\) by using transformations if needed.

Short answer

The graph of the function \(y = 1 - \frac{1}{2}{e^{ - x}}\) is observed to be increasing.

Step-by-step solution

Given data

The function is\(y = 1 - \frac{1}{2}{e^{ - x}}\).

Concept of Vertical and horizontal shifts

Vertical and horizontal shifts:

When\(y = f(x) + c\)and\(c > 0\), shift the graph of\(y = f(x)\)a distance\(c\)units upwards.

When\(y = f(x) - c\)and\(c > 0\), shift the graph of\(y = f(x)\)a distance\(c\)units downwards.

When\(y = f(x) - c\)and\(c > 0\), shift the graph of\(y = f(x)\)a distance\(c\)units towards the right.

When\(y = f(x) + c\)and\(c > 0\), shift the graph of\(y = f(x)\)a distance\(c\)units towards the left.

Sketch the graph of the function \(y =  - {e^x}\)

The standard graph of the function \(y = {e^x}\) is roughly sketched in figure 1 as follows:

Figure 1

Reflect the graph \(y = {e^x}\) about the \(y\)-axis and obtain the graph of \(y = {e^{ - x}}\).

Thus, the graph of \(y = {e^{ - x}}\) is drawn in figure 2 as shown below.

Figure 2

Sketch the graph of the function \(y = 1 - \frac{1}{2}{e^{ - x}}\)

Draw the graph of \(y = - {e^{ - x}}\), reflect the graph \(y = {e^{ - x}}\) about the \(x\)-axis in figure 3 as follows:

Figure 3

Here, figure 3 indicates the graph of \(y = - {e^{ - x}}\).

Then, draw the graph of \(y = - \frac{1}{2}{e^{ - x}}\), by shrunk the graph \(y = - {e^{ - x}}\) vertically by a factor of \(0.5\).

Figure 4

Here, figure 4 represents the graph of \(y = - \frac{1}{2}{e^{ - x}}\).

Then, obtain the graph of \(y = 1 - \frac{1}{2}{e^{ - x}}\) by shifting the graph of \(y = - \frac{1}{2}{e^{ - x}}\) one unit upward in figure 5 as shown below.

Figure 5

From figure 5, the graph of the function \(y = 1 - \frac{1}{2}{e^{ - x}}\) is observed to be increasing.