Question

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

Short answer

\(y = - {\left( {\frac{1}{2}} \right)^x}\) And \(y = {\left( {\frac{1}{2}} \right)^x}\) are observed to be reflecting in \(x\)-axis.

Step-by-step solution

Given data

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

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 \(y =  - {2^{ - x}}\)

Rewrite the given function as \(y = - {\left( {\frac{1}{2}} \right)^x}\).

The standard graph of the function \(y = {\left( {\frac{1}{2}} \right)^x}\) is roughly sketched in figure 1 as follows:

Figure 1

From figure 1, the graph of \(y = {\left( {\frac{1}{2}} \right)^x}\) is observed to be decreasing.Then, to draw the graph of \(y = - {\left( {\frac{1}{2}} \right)^x}\), reflect the graph \(y = {\left( {\frac{1}{2}} \right)^x}\) about the \(x\)-axis. Thus, the graph of \(y = - {\left( {\frac{1}{2}} \right)^x}\) is shown below in figure 2 as follows:

Figure 2

From figure 2, it is observed that \(y = - {\left( {\frac{1}{2}} \right)^x}\) and \(y = {\left( {\frac{1}{2}} \right)^x}\) are observed to be reflecting in \(x\)-axis.