Graphs of Functions

5.3 Special Exponential and Logarithmic Functions

Special Exponential Functions

The exponential function $y=e^x$ is a special exponential function, and is vital in the field of mathematics.
In particular, $e=\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}$ is a constant known as the Euler’s number. It is approximately 2.718281459045…, and it is an irrational number like $\pi$ .

Transformations with a Single Parameter

In previous section (A1 – Functions and Graphs/Transformation of Graphs), we have looked at transformations such as dilation, translation and reflection applied to specific functions and graphs.

We can expand this knowledge to transformations under a single, common parameter.

For instance, a translation of k units rightwards, and l units upwards.

In exact, we will look into the common characteristics that transformations of the same type under a unknown parameter share.

Circles

A picture containing chart Description automatically generated

The equation of a ‘standard’ circle is $x^{2}+y^{2}=r^{2}$ .
This circle has a radius of $r$ , and its centre is the origin $(0,\ 0)$ .

E.g. The right shows the graph of the circle with equation $x^{2}+y^{2}=4$ .

Note: Be careful here that the radius is 2, and not 4 as it should be $r^{2}=4,\ r=2$ .

By definition $r>0$ as it signifies the radius of the circle. However, if we observe the equation, $r$ can be negative too (the interpretation of the negative sign will involve a concept called ‘vectors’).
As we can see, $x$ and $y$ both have a range of $[-r,\ r]$ .

Graphs of Sine and Cosine Functions

The graph of where $-\pi<x<3\pi$ are plotted below. Do note that it extends beyond the drawn range.

There are a few observations we can make, and we can tie it back to what we have already learnt:

i) The graph repeats itself after an interval of $2\pi$ units. We say that the graph has a period of $2\pi$ , and hence is called a periodic function. Previously, we have learnt that $\sin(x+2\pi)$ .

The Logarithmic Function

The logarithmic function with base is defined as follows:

$a^{x} = b$ is equivalent to $log_{a}{b}=x$ , where $a \in R^{+}\setminus \left \{ 1 \right \}$ .

Note: ‘ $a \in R^{+}\setminus \left \{ 1 \right \}$ ‘ means that could be any positive number, excluding 1.

The expression $log_{a}{y}=x$ is read as: ‘the logarithm of $y$ to the base $a$ is equal to $x$ ’.

Example

The Exponential Function

The exponential function is defined as follows:

$y=a^x$ , where $a \in R^+\backslash {1}$ .

Note: ‘ $a \in R^+\backslash {1}$ ’ means that could be any positive number, excluding 1.

Example

$9=3^{2}=3 \times 3$ is equivalent to $log_{3}9=2$ .

$8=16^{\frac{3}{4}}=\sqrt[4]{16^{3}}=\sqrt{\sqrt{16 \times 16 \times 16}}$ .

Graph & Properties

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Graphs of Functions

5.3 Special Exponential and Logarithmic Functions

Special Exponential Functions

4.1 Transformations with a Single Parameter

Transformations with a Single Parameter

1.7 Additional Graph Types

Circles

1.6 Trigonometric Functions

Graphs of Sine and Cosine Functions

1.4 Logarithmic Functions

The Logarithmic Function

1.3 Exponential Functions [Free]

The Exponential Function

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