Transformation of equations

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

  • 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.