Question

To determine

(a) To sketch the function \(y = {\log _{10}}(x + 5)\) using the graphs from the given figures, use the concept of transformations if needed.

(b) To sketch the function \(y = - \ln x\) using the graphs from Figure 12 and Figure 13; use the concept of transformations if needed.

Short answer

(a) It is shifted \(5\) units to the left.

(b) It is reflected about the \(x\)-axis.

Step-by-step solution

Given data

The given functions are \(y = {\log _{10}}(x + 5)\) and \(y = - \ln x\).

Concept of Horizontal shift

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

Plot the graph of the function \(y = {\log _{10}}x\)

(a)

The graph of the function \(y = {\log _{10}}x\) is shown below in Figure 1.

Then, draw the graph of \(y = {\log _{10}}(x + 5)\), by shifting the graph \(y = {\log _{10}}x\) five units to the left.

Thus, the graph of \(y = {\log _{10}}(x + 5)\) is shown below in Figure 2.

Observe that above figure is obtained from the figure 1 in such a way that it is shifted 5 units to the left.

Plot of the graph of function \(y = \ln x\)

(b)

The graph of the function \(y = \ln x\) is shown below in Figure 3.

Then, draw the graph of \(y = - \ln x\), by reflecting the graph \(y = \ln x\) about \(x\)-axis.

Thus, the graph of \(y = - \ln x\) is shown below in Figure 4.

Observe that Figure 4 is obtained from Figure 3 in such a way that it is reflected about the \(x\)-axis.