Transformations of Graphs of Logarithmic Functions

Horizontal Shifts of the Parent Function $f(x)=log_b(x)$

For any constant $c$,  the function $f(x)=log_b(x+c)$

• shifts the parent function $y=log_b(x)$ left $c$ units if $c > 0$.
• shifts the parent function $y=log_b(x)$ right $c$ units if $c < 0$.
• has the vertical asymptote  $x=−c$.
• has domain $(−c, \infty)$.
• has range $(−\infty, \infty)$.

HOW TO

Given a logarithmic function with the form $f(x)=log_b(x+c)$, graph the translation.

1. Identify the horizontal shift:
   1. If $c > 0$,  shift the graph of $f(x)=log_b(x)$ left $c$ units.
   2. If $c < 0$,  shift the graph of $f(x)=log_b(x)$ right $c$ units.
2. Draw the vertical asymptote  $x=−c$.
3. Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting $c$ from the $x$ coordinate.
4. Label the three points.
5. The Domain is $(−c, \infty)$, the range is $(−\infty, \infty)$, and the vertical asymptote is $x=−c$.

Example 4

Graphing a Horizontal Shift of the Parent Function $y = log_b(x)$

Sketch the horizontal shift $f(x)=log_3(x−2)$ alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

Solution

Since the function is $f(x)=log_3(x−2)$, we notice $x+(−2)=x–2$.

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, $(\frac{1}{3}, −1), (1, 0)$, and $(3, 1)$.

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

Label the points $(\frac{7}{3}, −1), (3,0)$, and $(5,1)$.

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

Figure 7

Try It #4

Sketch a graph of $f(x)=log_3(x+4)$ alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.