Question

Make a rough sketch by hand of the graph of each function. Use the graphs given in Figures 12 and 13 and. If necessary, the transformations of section 1.3.

(a) \(y = {\bf{lo}}{{\bf{g}}_{{\bf{10}}}}\left( {x + {\bf{5}}} \right)\) (b) \(y = - {\bf{ln}}x\)

Short answer

(a) The graph is shown below:

(b) The graph is shown below:

Step-by-step solution

Write the transformation to find the graph of curve \(y = {\bf{lo}}{{\bf{g}}_{{\bf{10}}}}\left( {x + {\bf{5}}} \right)\)

The graph of \({\log _{10}}\left( {x + 5} \right)\) is obtained from the graph of \({\log _{10}}x\) by shifting it five units to the left.

Sketch the graph of \(y = {\bf{lo}}{{\bf{g}}_{{\bf{10}}}}\left( {x + {\bf{5}}} \right)\)

The figure below represents the graph of \({\log _{10}}x\) as shown below

The figure below represents the graph of \({\log _{10}}\left( {x + 5} \right)\) as shown below:

Write the transformation to find the graph of the curve \(y =  - {\bf{ln}}x\)

Reflect the graph of \(\ln x\) about x-axis to obtain the graph of \( - \ln x\).

Sketch the graph of \(y =  - {\bf{ln}}x\)

The figure below represents the sketch \(y = \ln x\).

The figure below represents the sketch of \(y = - \ln x\) as shown below:

Thus, the graphs are obtained.