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How Do You Write an Equation for a Translation of an Absolute Value Function?

Rewrite the equation y = 2|x-3|+4 to translate the graph to the right 1 and down 3.

Summary

  1. The form of an absolute value equation is y = a|x-h|+k
  2. 'a' is a constant that controls how much the graph is stretched or squeezed, and whether the graph opens upwards or downwards
  3. 'h' is a constant that controls the left and right translation
  4. 'k' is a constant that controls the up and down translation
  5. The vertex is where the two rays of the absolute value graph start and is at the point (h,k)
  6. Since 'a' is 2 in our equation, the graph is steeper than the graph of the parent function y = |x|
  7. In a translation the graph is the same size and shape as the original, just shifted to a new location

Notes

    1. The form of an absolute value equation is y = a|x-h|+k
    2. 'a', 'h', and 'k' are constants
    1. 'a', 'h', and 'k' are constants
    2. 'a' controls how much the graph is stretched or shrunk, and whether the graph opens upwards or downwards
    3. 'h' controls the left and right translation
    4. 'k' controls the up and down translation
    1. The vertex is where the two rays of the absolute value graph start
    2. 'a', 'h', and 'k' are constants
    3. 'a' controls how much the graph is stretched or shrunk, and whether the graph opens upwards or downwards
    4. 'h' controls the left and right translation
    5. 'k' controls the up and down translation
    1. 'h' controls the left and right translation
    2. 'k' controls the up and down translation
    1. Notice how there is a minus sign in front of 'h' in the general form
    2. That means when we pick out 'h', it will have the opposite sign of what's in front of it in the equation
    3. So since there is a minus sign in front of 3 in x-3, our value for 'h' will be a positive 3
    1. Since in the general form there is a plus sign in front of 'k', it will have the same sign that's in front of it in the equation when we pull it out
    2. So since we have a +4 at the end, 'k' will be positive 4
    3. Since we know that 'h' is 3 and 'k' is 4, our vertex (h,k) is the point (3,4)
    1. A horizontal translation means we're shifting the graph to the right or left
    2. In an absolute value equation, 'h' controls the left and right translation
    1. Remember, 'h' controls the left and right shift of an absolute value equation
    2. So to move the graph right one unit, we need to take the 'h' value we already have and add 1 to it
    3. The new 'h' value will be 3+1, or 4 once we simplify
    4. There's a minus sign before 'h' in the general equation, so be sure to put 3+1 in parentheses
    5. Our new equation, y = 2|x-4|+4, represents the graph of our original equation shifted to the right one unit
    1. A vertical translation means we're shifting the graph up or down
    2. In an absolute value equation, 'k' controls the up and down translation
    1. 'k' controls the up and down translation
    2. Any time k-values get smaller, the graph moves downwards
    3. When we subtract 3 from 4, we end up with a +1 on the end of our equation
    4. This gives us a new equation: y = 2|x-4|+1
    5. This new equation represents our original function shifted to the right 1 unit and down 3 units
    1. y = 2|x-4|+1 is the equation after being translated right 1 unit and down 3 units
    1. Remember, the vertex of an absolute value function is the point (h,k)
    2. The x-value of the vertex is the h-value of the new equation, 4
    3. The y-value of the vertex is the k-value of the new equation, 1
    4. Notice on the graph that our new vertex moved to the right one unit and down three units, just like our function was supposed to
    5. Our new graph is the same size and shape as the original one, just shifted to a new location in the coordinate plane