## 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.

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

\(q>0\) | ||

\(q<0\) |

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

\(q>0\) | ||

\(q<0\) |