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How Do You Graph a Translation of a Function?

Given the graph of f(x), graph the function f(x – 3) + 2

Summary

  1. The function f(x-3)+2 has the form f(x-h)+k, which represents a translation of a function
  2. A translation is a horizontal or vertical slide to a new location in the coordinate plane
  3. In a translation, the size and shape of the function stay the same
  4. 'h' represents the horizontal translation
  5. Positive 'h' values mean the graph will move to the right, and negative 'h' values mean it will move to the left
  6. 'k' represents the vertical translation
  7. Positive 'k' values mean the graph will move up, and negative 'k' values mean it will move down
  8. Since 'h' is a positive 3, the graph will be translated 3 units to the right
  9. Since 'k' is a positive 2, the graph will be translated 2 units up

Notes

    1. A graph of the function f(x) has been provided for you
    1. By looking at the graph of the function f(x), you can see that it spans from x = 2 to x = 7
    1. A translated function will be in the form f(x-h)+k, where 'h' and 'k' are constants
    2. Here we can see that f(x-3)+2 looks a lot like f(x-h)+k!
    1. We can have either a horizontal or vertical translation, or a combination of both
    2. Horizontal translations move the graph left or right
    3. Vertical translations move the graph up or down
    1. The problem asks you to translate the graph of f(x) to the graph of f(x-3)+2
    2. 'h' represents the number of horizontal units that the graph will shift
    3. 'k' represents the number of vertical units that the graph will shift
    1. Positive 'h' values will shift the graph to the right
    2. Negative 'h' values will shift the graph to the left
    1. Positive 'k' values will shift the graph up
    2. Negative 'k' values will shift the graph down
    1. This is the horizontal translation!
    2. Don't be fooled by the built-in minus sign before 'h': 'h' is actually a positive 3 here!
    1. 'h' represents the number of units the graph will shift horizontally
    2. Positive 'h' values shift the graph to the right and negative 'h' values shift the graph to the left
    1. This is the vertical translation!
    1. 'k' represents the number of units the graph will shift vertically
    2. Positive 'k' values shift the graph up and negative 'k' values shift the graph down
    1. We know now that the new graph should be shifted 3 units to the right and 2 units up
    1. We know now that the new graph should be shifted 3 units to the right and 2 units up
    2. So the point (2,0) will translate to (2+3,0+2), or (5,2)
    3. The point (3,2) will translate to (3+3,2+2), or (6,4)
    4. The point (5,3) will translate to (5+3,3+2), or (8,5)
    5. The point (7,3) will translate to (7+3,3+2), or (10,5)
    6. Connecting the new points will give you the new graph!
    1. Remember, if we have an 'h' term, that means there will be a horizontal translation
    2. If we have a 'k' term, that means there will be a vertical translation