Question

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

Short answer

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

Step-by-step solution

Given data

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

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 as shown in figure 1.

Figure 1

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

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

Figure 2

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

Draw the graph of \(y = 1 - {e^x}\), shift the graph \(y = - {e^x}\) one unit upward.

Thus, the graph of \(y = 1 - {e^x}\) is shown in figure 3.

Figure 3

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

Thus, the graph of \(y = 2\left( {1 - {e^x}} \right)\) is shown in figure 4.

Figure 4

From figure 4, the graph of the function \(y = 2\left( {1 - {e^x}} \right)\) is observed to be decreasing.