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Transformation of Functions

4.3 Using Matrices for Transformations and its Inverse

Using Matrices for Transformations

  • Some transformations can be represented as matrices. For instance, all linear transformations can be shown by matrices.

Note: Linear transformation is a general type of transformation that will not be studied in this course. Reflection and dilation are actually particular types of linear transformation, while translation is not.

  • Generally, linear transformations (which includes reflection and translation) can be described as:

\left[\begin{array}{l} x \\y\end{array}\right] \rightarrow\left[\begin{array}{ll}a & b \\c & d\end{array}\right]\left[\begin{array}{l}x \\y\end{array}\right]=\left[\begin{array}{l}a x+b y \\c x+d y\end{array}\right]

which describes the rule (x,\ y) \rightarrow(a x+b y,\ c x+d y).

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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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3.9 Inverse Transformations

Note: In this note, we will only cover the inverse for translation(s) and dilation(s).

Inverse Transformations

  • A generic way of writing any combination of translation and dilations (which already includes reflections) is

y=A f(n(x+b))+c

where A,\ n,\ b,\ c \in R,\ n \neq 0.

In this transformation, we have (x,\ y) \rightarrow\left(\frac{x-b}{n},\ A y+c\right).

A simple idea is: Try to cancel out the effects on x. So x^{\prime}=\frac{x-b}{n}. As we originally have y=f(x), so f(n(x+b))=y if we adjust x to cancel n and b. So the new , here labelled as y^{\prime} to avoid confusion, is y^{\prime}=A y+c.

And finally (x,\ y) \rightarrow\left(x^{\prime},\ y^{\prime}\right) and sub x^{\prime},\ y^{\prime} to get the expression above. This is covered in 3.4 Combination of Transformations in more detail.

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3.8 Transformations of Trigonometric Functions

Transformations with Sine and Cosine Graphs

  • We can see that the graph of y=\cos x can simply be obtained by y=\sin x shifted \frac{\pi}{2} to the left. This is a translation, and the identity is not unfamiliar to us: \sin x=\cos \left(x-\frac{\pi}{2}\right).
  • Now we shall look at graphs of simple transformations of these graphs. We shall start with y=a \sin (n t) and y=b \cos (n t). Naturally, n would be a dilation from the y-axis, and a (and b) is a dilation from the x-axis. The mechanism is similar, and is explained in A1 – Functions and Graphs/Relationship of Transformations . Here we will only provide a few simple examples.
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3.5 Transformations of Polynomial Functions

Transformations with Power Functions

  • Recall that power functions has the form y=x^n, where is a real number. We have looked into cases where is a rational number, which includes positive and negative whole numbers and fractions.
  • Transformation can be applied on power functions. The transformed equation simply take the general form of f(x)=a(b(x-c))^{n}+d, where a,\ b,\ c,\ d are real numbers.

i) If it is a reflection, we have b=-1 or a=-1.

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3.3 Transformation 3: Reflection

Reflection

  • Reflection is a more familiar transformation to us: we usually see our reflections in a mirror.
  • In Cartesian plane, a line will be the ‘mirror’, and the function will be reflected along the ‘mirror’. Therefore, the line can be horizontal, vertical, or diagonal. Here, we will only consider horizontal and vertical reflection, as diagonal reflections are too complex for this course. In particular, we will only study reflections along the x-axis and y-axis.
  • A reflection along the x-axis gives the following transformation:

(x,\ y) \rightarrow(x,\ -y)

y=-f(x)

and a reflection along the y-axis gives the following transformation:

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3.2 Transformation 2: Dilation

Dilation

  • Dilation is a type of transformation. A translation is simply a ‘stretch’ or a ‘shrink’. The direction is important too, hence it can be a stretch along the x- and/or y-axis.
  • ‘A dilation of factor h from the x-axis and of factor k from the y-axis’, where h and k are positive numbers, is described by the rule

(x,\ y) \rightarrow(k x,\ h y)

Note: Notice the positions of and and their corresponding axis in the description.

  • If a function initially is y=f(x), the new function under the translation as described above is
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