Transformations of Graphs of Logarithmic Functions

Vertical Stretches and Compressions of the Parent Function $y=log_b(x)$

For any constant $a > 1$, the function $f(x)=alog_b(x)$

• stretches the parent function $y=log_b(x)$ vertically by a factor of $a$ if $a > 1$.
  
• compresses the parent function $y=log_b(x)$ vertically by a factor of $a$ if $0 < a < 1$.
  
• has the vertical asymptote $x=0$.
  
• has the $x$-intercept $(1,0)$.
  
• has domain $(0, \infty)$.
  
• has range $(−\infty, \infty)$.
  

HOW TO

Given a logarithmic function with the form $f(x)=alog_b(x), a > 0$, graph the translation.

1. Identify the vertical stretch or compressions:
   
   • If $|a| > 1$, the graph of $f(x)=log_b(x)$ is stretched by a factor of $a$ units.
     
   • If $|a| < 1$, the graph of $f(x)=log_b(x)$ is compressed by a factor of $a$ units.
     
2. Draw the vertical asymptote $x=0$.
   
3. Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying the $y$ coordinates by $a$.
   
4. Label the three points.
   
5. The domain is $(0, \infty)$, the range is $(−\infty, \infty)$, and the vertical asymptote is $x=0$.
   

Example 6

Graphing a Stretch or Compression of the Parent Function $y = log_b(x)$

Sketch a graph of $f(x)=2log_4(x)$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Solution

Since the function is $f(x)=2log_4(x)$, we will notice $a=2$.

This means we will stretch the function $f(x)=log_4(x)$ by a factor of 2.

The vertical asymptote is $x=0$.

Consider the three key points from the parent function, $(\frac{1}{4},−1), (1,0)$, and $(4,1)$.

The new coordinates are found by multiplying the $y$ coordinates by 2.

Label the points $(\frac{1}{4},−2)$, (1,0)\), and $(4,2)$.

The domain is $(0, \infty)$, the range is $(−\infty, \infty)$, and the vertical asymptote is $x=0$. See Figure 11.

Figure 11

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

Try It #6

Sketch a graph of $f(x)=\frac{1}{2}log_4(x)$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Example 7

Combining a Shift and a Stretch

Sketch a graph of $f(x)=5log(x+2)$. State the domain, range, and asymptote.

Solution

Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5, as in Figure 12. The vertical asymptote will be shifted to $x=−2$. The $x$-intercept will be $(−1,0)$. The domain will be $(−2, \infty)$. Two points will help give the shape of the graph: $(−1,0)$ and $(8,5)$. We chose $x=8$ as the $x$-coordinate of one point to graph because when $x=8, x+2=10$, the base of the common logarithm.

Figure 12

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

Try It #7

Sketch a graph of the function $f(x)=3log(x−2)+1$. State the domain, range, and asymptote.