Question

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

Short answer

The \(y\)-intercept of the graph of \(y = {e^{|x|}}\) is \(1\).

Step-by-step solution

Given data

The function is\(y = {e^{|x|}}\).

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 \(y\)-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

Then, to draw the graph of \(y = {e^{|x|}}\), remove the portions of the graphs that appear below \(y = 1\) in figure 1 and figure 2.

Thus, the graph of \(y = {e^{|x|}}\) is obtained in figure 3 as shown below.

Figure 3

From figure 1, it is observed that the \(y\)-intercept of the graph of \(y = {e^{|x|}}\) is \(1\).