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Graphs of Functions

4.1 Transformations with a Single Parameter

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.
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1.7 Additional Graph Types


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  • 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].
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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).

Read More »1.6 Trigonometric Functions

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’.


Read More »1.4 Logarithmic Functions

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.


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]