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How Do You Translate a Function?

How do you translate a function?

Summary

  1. To horizontally translate a function, substitute 'x-h' for 'x' in the function
  2. The value for 'h' controls how much the graph shifts to the left or right
  3. In our example, since h = -4, the graph shifts 4 units to the left
  4. To vertically translate a function, add 'k' onto the end
  5. The value for 'k' controls how much the graph shifts up or down
  6. In our example, since k = -5, the graph shifts 5 units down
  7. You can also perform both horizontal and vertical translations on a function at the same time!

Notes

    1. A horizontal translation moves the graph left or right
    2. A vertical translation moves the graph up or down
    1. A horizontal translation moves the graph left or right
    1. 'x' represents the x-value of the function
    2. 'h' is the number of units that the function will move to the left or right
    1. 'h' is the number of units that the function will move to the left or right
    2. The '>' symbol means 'greater than', so 'h > 0' is the same as saying 'when h is positive'
    3. So the graph shifts right when 'h' is positive
    1. 'h' is the number of units that the function will move to the left or right
    2. The '<' symbol means 'less than', so 'h < 0' is the same as saying 'when h is negative'
    3. So the graph shifts left when 'h' is negative
    1. We're going to horizontally translate the graph for f(x) = |x|
    2. |x| means 'the absolute value of x', so this is an absolute value function
    1. 'h' is the number of units that the function will move to the left or right
    2. Remember, a negative 'h' will translate the function to the left
    3. So an 'h' value of -4 will translate the function 4 units to the left
    1. 'x' represents the x-value of the function
    2. 'h' represents the number of units that the function will be horizontally translated
    3. Recall that our original function was f(x) = |x|, so replacing 'x' with 'x-h' will give you f(x-h) = |x-h|
    1. 'h' represents the number of units that the function will be horizontally translated
    2. Recall that a negative 'h' will translate the function to the left
    3. Don't be fooled by the positive 4: the function will still translate 4 units to the left because 'h' is -4
    1. The new function has the form f(x+4) = |x+4|
    2. Don't be fooled by the positive 4: the function will still translate 4 units to the left because 'h' is -4
    1. A vertical translation moves the graph up or down
    1. 'x' represents the x-value of the function
    2. Notice here that by adding 'k' onto the end, we're not actually changing 'x' in the function
    3. Since we're shifting up or down, only the y-value will change
    4. 'k' represents the number of units that the function will be vertically translated
    1. 'k' represents the number of units that the function will be vertically translated
    2. The '>' symbol means 'greater than', so 'k > 0' is the same as saying 'when k is positive'
    3. So the graph shifts up when 'k' is positive
    1. 'k' represents the number of units that the function will be vertically translated
    2. The '<' symbol means 'less than', so 'k < 0' is the same as saying 'when k is negative'
    3. So the graph shifts down when 'k' is negative
    1. We're going to vertically translate the graph for f(x) = √x
    2. f(x) = √x is a square root function, since it has 'x' under a square root
    1. 'k' represents the number of units that the function will be vertically translated
    2. Remember, a negative 'k' will translate the function down
    3. So since 'k' is -5, the function will shift down 5 units
    1. 'k' represents the number of units that the function will move up or down
    1. Since 'k' is -5, we can just plug that value in for 'k'
    1. Since our value for 'k' was negative, our graph shifted down 5 units
    2. 'k' represents the number of units that the function moves up or down
    1. We can also do both horizontal and vertical translations on a function at the same time
    1. We're going to horizontally and vertically translate the graph for f(x) = x2
    2. f(x) = x2 is a quadratic function, since its highest degree term is an x2
    1. 'h' represents the number of units that the function will be translated to the left or right
    2. Positive values for 'h' will translate the function to the right
    3. 'k' represents the number of units that the function will be translated up or down
    4. Positive values for 'k' will translate the function up
    1. 'h' represents the number of units that the function will be translated to the left or right
    2. Note that if 'h' was -3, the graph would shift three units to the left
    1. 'k' represents the number of units that the function will be translated up or down
    2. Note that if 'k' was -4, the graph would shift four units down
    1. 'h' represents the number of units that the function will be horizontally translated
    2. 'k' represents the number of units that the function will be vertically translated
    1. Since 'h' is 3, that means our function will be translated 3 units to the right
    2. Since 'k' is 4, that means our function will be translated 4 units up
    1. So it is possible to perform horizontal and vertical translations together!
    2. 'h' will determine how much the graph shifts to the left or right
    3. 'k' will determine how much the graph shifts up or down