Question

Sketch the graph of the function \(y = {(0.5)^x}\) by using transformations if needed.

Short answer

The \(y\)-intercept of \(y = {(0.5)^x} - 2\) is \((0, - 1)\).

Step-by-step solution

Given data

The function is\(y = {(0.5)^x}\).

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 \(y = {(0.5)^x}\)

The standard graph of the function \(y = {(0.5)^x}\) is roughly sketched in figure 1 as follows:

Figure 1

Then, to draw the graph of \(y = {(0.5)^x} - 2\), shift the entire graph of \(y = {(0.5)^x}\) two unit downwards.

Thus, the graph of \(y = {(0.5)^x} - 2\) is shown below in figure 2 as follows:

Figure 2

From figure 2, it is observed that the \(y\)-intercept of \(y = {(0.5)^x} - 2\) is \((0, - 1)\).