Now, let’s study the effects of changing the parameter c in our exponential form

f(x) = a * log_B(x + c) + d.

Log Function Applet

Slowly decrease the value of c from its initial value of zero by moving the third slider to the left, causing it to become negative. Now, move the slider back to the right until c has become positive.

• Do you observe any changes in the range? If so, describe the changes.
  Interactive popup. Assistance may be required.

  Check Your Answer

  No. The range remain the same regardless of c.
  
  
• Is there a change in the domain? If so, describe the changes.
  Interactive popup. Assistance may be required.

  Check Your Answer

  Yes. The domain starts at the opposite of c, or -c. For example, if c = 4, the domain would be (-4,∞); if c = -2, the domain would be (2,∞). This c constant causes a horizontal shift to occur.
  
  
• Is there a change in the vertical asymptote? If so, where?
  Interactive popup. Assistance may be required.

  Check Your Answer

  Yes. The vertical asymptote always occurs at the opposite of c, or -c. For instance, if c = 2.6, the vertical asymptote would occur at c = -2.6.
  
  
• Return the slider to its original value of c = 0. First, locate the point (1,0) on the graph, marked by the red dot. Now, move the fourth slider and change the d-value to d = 3. Notice the function no longer passes through (1,0); however, due to the vertical shifting caused by d, the point (1,3) is now part of the function.
  
  Does vertical shifting caused by d alter the domain, range or vertical asymptote?
  Interactive popup. Assistance may be required.

  Check Your Answer

  No. The domain, range, and vertical asymptote all remain the same.

This study is very significant to us. It turns out the domain, range, and vertical asymptote caused by changing a, B, and c can be summed up as follows:

• There will be a vertical asymptote at x = -c for any given c-value.
• The domain will start or stop at -c, as (-c,∞ ).
• The range is always (-∞,∞) for all basic logarithmic functions of the form f(x) = a * log_B(x + c) + d, regardless of a, B, c, or d.

Go ahead and test it now. Set a value of c = 5. Now, explore changing a, B and d by moving the sliders. Does the vertical asymptote remain at x = -5? Does the domain have a limiting edge of -5 also? The domain should be (-5,∞). Is your range (-∞,∞) regardless of any changes? Amazing!