Transformations of functions: Horizontal translations

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Examples

Lessons

    1. Sketch the following functions on the same set of coordinate axes:
      y=(x)2y = {\left( x \right)^2}      VS.      y=(x6)2y = {\left( {x - 6} \right)^2}      VS.      y=(x+5)2y = {\left( {x + 5} \right)^2}
    2. Compared to the graph of y=x2y = {x^2}:
      • the graph of y=(x6)2y = {\left( {x - 6} \right)^2} is translated "horizontally" ________ units to the ______________.
      • the graph of y=(x+5)2y = {\left( {x + 5} \right)^2} is translated "horizontally" ________ units to the ______________.
  1. Horizontal Translations
    Given the graph of y=f(x)y = f\left( x \right) as shown, sketch:
    1. y=f(x8)y = f\left( {x-8} \right)
    2. y=f(x+3)y = f\left( {x+3} \right)
    3. In conclusion:
      (x)(x8)\left( x \right) \to \left( {x-8} \right): shift __________ to the __________. All x coordinates \Rightarrow ____________________
      (x)(x+3)\left( x \right) \to \left( {x+3} \right): shift __________ to the __________. All x coordinates \Rightarrow ____________________
      Vertical translations

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Topic Basics
Horizontal translations refer to movements of a graph of a function horizontally along the x-axis by changing the x values. So, if y = f(x), then y = (x –h) results in a horizontal shift. If h > 0, then the graph shifts h units to the right; while If h < 0, then the graph shifts h units to the left.
Compared to y=f(x)y=f(x):
y=f(x8)y=f(x-8): shift 8 units to the right
y=f(x+3)y=f(x+3): shift 3 units to the left