Maria Meringolo

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How can we recognize transformations of a function?

Using the TI & GraphCalc

 

 

Just incase you forgot…

 

Vertical and Horizontal Shifts (Translations)

 

Let c be a positive real number, and let f(x) be a function

 

y = f(x) + c       Shifted Upward                        (added c to the entire function)

y = f(x) – c       Shifted Downward                    (subtracted c from the entire function)

y = f(x + c)       Shifted Left                               (added c directly to the x)

y = f(x – c)       Shifted Right                             (subtracted c directly from the x)

 

 

Reflections

 

Given f(x) is a function

 

y = -f(x)           r_x-axis                                      (the entire function is negated)

y = f(-x)           r_y-axis                             (only the x is negated)

 

 

Original function y = x²

(0, 0)

 

 

 

              y = x² – 4                                                                           y = x² – 2

(0, -2)

(0, -4)

                                   

         moved down 4 units                                                         moved down 2 units

 

 

 

 

                 y = x² + 5                                                                        y = (x – 4)²

(4, 0)

(0, 5)

                                   

          moved up 5 units                                                        moved to the right 4 units

 

 

 

 

               y = (x – 2)²                                                                                            y = (x + 6)²

(-6, 0)

(2, 0)

                                     

     moved to the right 2 units                                                     moved to the left 6 units

A few more examples…

 

          Graph y = √x                                    then                             Graph y = - √x

                                          

 

What transformation took place by negating the function?

 

        Graph y = √ - x

 

What transformation took place by negating just the x?

 

1. Describe the transformation of each graph when compared to y = │x│:

 

a)      y = -│x│                                 Reflection over the x - axis

 

b)      y = │x + 2│                             Shifted to the left 2 units

 

c)      y = │x + 4│- 8                        Shifted to the left 4 units and down 8 units

 

d)      y =│- x│                                 Reflection over the y – axis

 

 

2. Write an equation that will translate y = x²:

 

a)      5 units right and 3 units down                                                    y = (x – 5)² - 3

b)      reflection over the x-axis and move 4 units right             y = -(x – 4)²