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# Asymptotes

## 3.7 Transformations of Logarithmic Functions

#### Plotting Logarithmic Functions

Example

Say y=log_{e}{x}. Ideally, you should remember the general shape of the graph, then labelling any important points (i.e. Step 2).

Generally, the steps to plot a graph are:

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## 3.6 Transformations of Exponential Functions

#### Plotting Exponential Functions

Example

Say y=2^{x}. Ideally, you should remember the general shape of the graph, then labelling any important points (i.e. Step 2). You should try yourself for y=0.5^{x}.

Generally, the steps to plot a graph are:

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## 1.6 Trigonometric Functions

#### 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).

## 1.4 Logarithmic Functions

#### 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

## 1.3 Exponential Functions [Free]

#### 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

Read More »1.3 Exponential Functions [Free]

## 1.2 Power Functions

#### Introduction to Power Functions

• Power functions are functions with the form f(x)=x^{n}, where n is a rational number.
• Some examples of power functions are:

f(x)=x^{2},\ f(x)=x^{4},\ f(x)=x^{\frac{1}{4}},\ f(x)=x^{-5},\ f(x)=x^{\frac{1}{3}}

• The expression x^n is read as: ‘x to the power of n’.

Drawing Power Functions (for Positive Integers) and its Graph Properties

## 1.1 Introduction to Graph Sketching

#### Introduction to Graph Sketching

• In this section, 3 types of functions will be introduced accordingly:

i) Power functions, y=x^n, where is a number, n\neq 0.

This will be further extended into polynomial functions, which has the general form y=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0}, where a_i are real numbers, a_n\neq 0.

ii) Exponential functions, y=a^x and also logarithm functions, y=log_ax.

iii) Trigonometric functions or circular functions, which includes y=\sin x,\ y=\cos x,\ y=\tan x and more.

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## Exponential and Logarithm Functions

This tutorial covers material encountered in chapter 5 of the VCE Mathematical Methods Textbook, namely: Exponential functions Index laws Log functions Log laws and change… Read More »Exponential and Logarithm Functions