HORIZONTAL AND VERTICAL TRANSLATIONS

Transcription

1 MCR3U Sections Transformations HORIZONTAL AND VERTICAL TRANSLATIONS A change made to a figure or a relation such that the figure or graph of the relation is shifted or changed in shape. Translations, reflections and stretches/compressions are types of transformations. * In order to sketch the graphs of the transformed functions, we first need to know how to graph the base function. Translations: A transformation that results in a shift of the original figure without changing its shape. 1) Vertical Translation of c units : * The graph of the function g(x) = f(x) + c. when c is positive, the translation is UP by c units. when c is negative, the translation is DOWN by c units. 2) Horizontal Translationof d units : * The graph of the function g(x) = f(x d). when d > 0, the translation is to the RIGHT by d units. when d < 0, the translation is to the LEFT by d units.

2 Example 1: Given the functions f(x) graphed below, sketch the graph of g(x) on the same grid as the original function f(x). a) g(x) = f(x) 5 b) g(x) = f(x + 3) Example 2: Graph the following parent functions and their transformed functions. Describe the transformations that occurred. State the domain and range of the transformed functions.

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4 REFLECTIONS OF FUNCTIONS Reflection : A transformation in which a figure is reflected over a reflection line. Invariant Points : Points that are unaltered by a transformation. 1) Reflection in the x axis or Vertical Reflection: The graph of g(x) = f(x) Examples: Graph the following functions along with their base functions. Describe the transformations that occurred. State the domain and range. State the equations of asymptotes, if any, and state any invariant points.

5 2) Reflection in the y axis or Horizontal Reflection: The graph of g(x) = f( x) Examples: Graph the following functions along with their base function. Describe the transformations that occurred. State the domain and range and equations of asymptotes, if any.

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7 Combine Reflections and Translations Examples: For the following graphs, describe the transformations of f(x). Write the equations of the transformed functions

8 Examples: Graph the following functions along with their parent functions. Describe the transformations that occurred. State the domain and range of the transformed functions and the equations of the asymptotes, if any.

9 STRETCHES OF FUNCTIONS Stretches and compressions are transformations that cause functions to change shape. 1) Vertical Stretch or Compression : The graph of the function g(x) = af(x), a 0 Examples: Graph the following functions along with their parent functions. Describe the transformations that occurred. State the domain and range of the transformed functions and the equations of any asymptotes.

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11 2) Horizontal Stretch or Compression : The graph of the function g(x) = f(kx), k 0 when k > 1, there is a HORIZONTAL COMPRESSION by a factor of when 0 < k < 1, there is a HORIZONTAL STRETCH/EXPANSION by a factor of Examples : Graph the following functions along with their parent functions. Describe the transformations that occurred. State the domain and range of the transformed functions and the equations of any asymptotes. State any invariant points.

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13 COMBINING TRANSFORMATIONS OF FUNCTIONS * To perform combinations of transformations, note. Stretches, compressions and reflections can be performed in any order as long as they are performed before translations. The function must be written in the form y = af [k(x d)]+ c to identify the specific transformations. Examples: Graph the following pairs of functions on the same grid. Describe the transformations that occured. State the domain and range of each transformed function and the equations of any asymptotes.

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