Revision of Exponential Functions

Functions of the form \(y=a{b}^{x}+q\)

Functions of the general form \(y=a{b}^{x}+q\), for \(b>0\), are called exponential functions, where \(a\), \(b\) and \(q\) are constants.

The effects of \(a\), \(b\) and \(q\) on \(f(x) = ab^x + q\):

• The effect of \(q\) on vertical shift

  • For \(q>0\), \(f(x)\) is shifted vertically upwards by \(q\) units.

  • For \(q<0\), \(f(x)\) is shifted vertically downwards by \(q\) units.

  • The horizontal asymptote is the line \(y = q\).

• The effect of \(a\) on shape

  • For \(a>0\), \(f(x)\) is increasing.

  • For \(a<0\), \(f(x)\) is decreasing. The graph is reflected about the horizontal asymptote.

• The effect of \(b\) on direction

  Assuming \(a > 0\):

  • If \(b > 1\), \(f(x)\) is an increasing function.
  • If \(0 < b < 1\), \(f(x)\) is a decreasing function.
  • If \(b \leq 0\), \(f(x)\) is not defined.

\(b>1\)	\(a<0\)	\(a>0\)
\(q>0\)
\(q<0\)

\(0	\(a<0\)	\(a>0\)
\(q>0\)
\(q<0\)