6.4 Graphs of logarithmic functions  (Page 3/8)

Before graphing, identify the behavior and key points for the graph.

• Since $b=5$ is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote $x=0,$ and the right tail will increase slowly without bound.
• The x -intercept is $\left(1,0\right).$
• The key point $\left(5,1\right)$ is on the graph.
• We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points (see [link] ).

The domain is $\left(0,\infty \right),$ the range is $\left(-\infty ,\infty \right),$ and the vertical asymptote is $x=0.$

Since the function is $f(x)={\log }_3(x-2),$ we notice $x+\left(-2\right)=x–2.$

Thus $c=-2,$ so $c<0.$ This means we will shift the function $f(x)={\log }_3(x)$ right 2 units.

The vertical asymptote is $x=-(-2)$ or $x=2.$

Consider the three key points from the parent function, $\left(\frac{1}{3},\mathrm{-1}\right),$ $\left(1,0\right),$ and $\left(3,1\right).$

The new coordinates are found by adding 2 to the $x$ coordinates.

Label the points $\left(\frac{7}{3},\mathrm{-1}\right),$ $\left(3,0\right),$ and $\left(5,1\right).$

The domain is $\left(2,\infty \right),$ the range is $\left(-\infty ,\infty \right),$ and the vertical asymptote is $x=2.$