Graphing Functions

    In this topic we show the graphs for the five basic functions: the square function, the abolute value function, the cube function, the square root function, and the cube root function. Translations are covered by shifting graphs vertically and horizontally. Basically, vertical shifts work the same as the given sign and horizontal shifts work opposite the given sign. Then reflections of these graphs are shown, as well as, examples with translations and reflections. Finally, vertical stretches and vertical shrinks are illustrated.

    The basic elementary functions and their graphs are:












    


    

Definition (Vertical Translations) If a new function is obtained by taking a given function and performing then we have a vertical translation of We say is "shifted up by units" when and "shifted down by units" when

Example (Vertical Translations) Use vertical tranlsations to graph the functions and



(vertical shift down 2 units)


  
(vertical shift down 1 units)


  
(vertical shift up 2 units)


  
(vertical shift down 5 units)

    
  
(vertical shift up 2 units)

    

Definition (Horizontal Translations) If a new function is obtained by taking a given function and performing then we have a horizontal translation of We say is "shifted left by units" when and "shifted right by units" when

Example (Horizontal Translations) Use horizontal tranlsations to graph the functions and



(horizontal shift right 2 units)



(horizontal shift right 1 units)


  
(horizontal shift left 2 units)


  
(horizontal shift right 5 units)

    
  
(horizontal shift left 2 units)

    

Example (Combining Vertical and Horizontal Translations) Use horizontal and vertical tranlsations to graph the functions and


(horizontal shift right 2 units, vertical shift down 1 unit)


  
(horizontal shift right 1 unit, vertical shift up 2 units)


  
(horizontal shift left 2 units, vertical shift down 3 units)


  
(horizontal shift right 5 units, vertical shift down 5 units)

    
  
(horizontal shift left 2 units, vertical shift up 4 units)

    

Definition (Reflections) If a new function is obtained by taking a given function and performing then we have a vertical reflection of We say is "reflected through the -axis".

Example (Reflections) The graphs of the basic elementary functions with a reflection are:












    


    

Example (Combining Translations and Reflections) Here are five examples of translations with reflections of the basic graphs:


(reflection and vertical shift down 3)


  
(reflection, horizontal shift right 2, and vertical shift up 1)


  
(reflection, horizontal shift left 2 units, vertical shift up 3 units)


  
(reflection, horizontal shift right 1 units, and vertical shift down units)

    
  
(reflection and vertical shift up 1 units)

    

Definition (Vertical Stretches and Shrinks) If a new function is obtained by taking a given function and performing then we have a vertical shrink of when and a vertical stretch of when

Example (Vertical Stretches and Shrinks) Here are five examples of translations, reflections, and vertical stretches/shrinks of the basic graphs. The graph that is dashed is before the vertical stretch/shrink.


(reflection, horizontal shift left 1, and vertical shift down 3, and vertical stretch by 2)


  
(reflection, horizontal shift right 1, vertical shift up 2, and vertical shrink by )


  
(reflection, horizontal shift left vertical shift down 5, and vertical shrink by )


  
(reflection, horizontal shift right 1, vertical shift up and vertical stretch by 2)

    
  
(reflection, horizontal shift left 1, vertical shift down 2, and vertical stretch by 3)

    

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