Now let's study the effects of changing the parameter c in our exponential form:

f(x) = a * B^{x - c} + d

Slowly decrease the value of c 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 domain, range, and horizontal asymptote ?

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Check Your Answer

No. They all remain the same regardless of c.

Return the slider to its original value of c = 0.

Move the fourth slider and change the d-value to d = -3. Again, the domain remains (-∞, ∞), but what important changes do you notice regarding the range and horizontal asymptote?

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Check Your Answer

The range is now (-3, ∞), and the horizontal asymptote is now y = -3.

This is very significant to us. It turns out the vertical shift caused by d always causes two very important changes, regardless of a, B, or c!

A) There will be a horizontal asymptote at y = d for any given d-value.

B) The range will start or stop at d, as either (d, ∞) or (-∞, d), depending on the values of a, c, and B.

We also have learned a third important note.

C) The domain is always (-∞, ∞) for all exponential functions, regardless of a, B, c, or d.

Go ahead and test it now. Set a value of d = 3. Now explore changing a, c, and B by moving the sliders up and down. Does the horizontal asymptote remain at y = 3? Does the range have a limiting edge of 3 also? The range should be (3, ∞) or (-∞, 3) depending on the values of a, c, and B. Is your domain (-∞, ∞)? Amazing!