# Transformation of Equations

## Linear Transformations

• Linear transformations involve adding, subtracting, multiplying or dividing both sides of an equation by the same amount.
• A simple example would be transforming the equation x = 2 to x - 1 = 2 - 1, thus giving x - 1 = 1.

• Quadratic transformations often involve rearranging equations into the form y = ax² + bx + c.
• For example, y = x² + 2x + 1 could be transformed to y = (x + 1)² for easier manipulation and graphing.

## Completing the Square

• Completing the square is a strategy used to transform a quadratic equation from standard form to vertex form.
• Vertex form is given by y = a(x - h)² + k, where (h, k) represents the vertex of the parabola.

## Cubic Transformations

• Cubic transformations involve changing the expression y = ax³ + bx² + cx + d.
• For example, a negative value of ‘a’ results in a vertically reflected transformation, while a value of ‘a’ greater than 1 or less than -1 gives a vertically stretched or contracted transformation.

## Rational Function Transformations

• Rational function transformations involve fractional expressions, where the numerator and/or denominator are polynomials.
• They often result in asymptotes, or lines that the graph of the function approaches but never touches.

## Exponential and Logarithmic Transformations

• Exponential transformations adjust the function y = a^x, while logarithmic transformations change the function y = log_b(x).
• For these transformations, remember the horizontal and vertical translations, flips, and stretching rules.

## Trigonometric Transformations

• Transformations of trigonometric functions include y = a sin(bx + c) + d and y = a cos(bx + c) + d where a, b, c and d each have different effects on the graph.
• Remember that amplitude is affected by ‘a’, period changes by ‘b’, horizontal shift by ‘c’, and vertical shift by ‘d’.

## Inverse Transformations

• Inverse transformations involve swapping the x and y values in the equation, which results in the reflection of the graph in the line y=x.
• The ‘inverse’ of a function can only exist if the function is one-to-one, i.e., each x-value corresponds to exactly one y-value.

## Transformations Involving Complex Numbers

• When involving complex numbers, transformations become more complex. The transformation z -> w where w = f(z) can create all sorts of interesting results, especially when visualised on the complex plane.
• Examples of complex transformations include translation, rotation, dilation and reflection. These involve different combinations of addition, multiplication, taking reciprocals and conjugates.

## Use of Graphing Tools

• Graphical tools or software can be a powerful way to explore and visualise the effects of mathematical transformations. Particularly for more complex transformations, graphing tool can help to quickly understand the impact and end result.