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What Does the Constant 'h' do in y = |x-h|?

What does the constant h do in y = |x-h|?

Summary

  1. The vertical bars around 'x-h' are the symbol for absolute value
  2. The graph of the parent function, y = |x|, is a 'V' shape with its vertex at the origin
  3. When 'h' is positive, the graph shifts to the right
  4. When 'h' is negative, the graph shifts to the left
  5. 'h' ends up being the x-coordinate of the vertex

Notes

    1. The vertical bars around 'x-h' are the symbol for absolute value
    2. y = |x| is the parent function of y = |x-h|
    3. It's the simplest form of an absolute value function
    1. The graph of the parent function, y = |x|, is a 'V' shape with its vertex at the origin
    1. Setting the constant 'h' equal to 1 gives us y = |x-1|
    2. The vertical bars around 'x-1' are the symbol for absolute value
    1. To graph the equation, we want to make a table of values to find some (x,y) ordered pairs to plot
    2. Let's choose -2, -1, 0, 1, and 2 as some values for 'x'
    3. Then we can plug each value for 'x' into y = |x-1| to get the y-values
    4. Remember, the absolute value of a negative number is just that same number as a positive
    5. So the absolute value of -2-1, or -3, is just 3
    6. The absolute value of -1-1, or -2, is 2, and the absolute value of 0-1, or -1, is 1
    1. Our new graph is the same shape and size as the original graph
    2. All we did was slide it right one unit
    3. Slides like this are called 'translations'
    4. Since our graph moved horizontally, we call this a 'horizontal translation'
    1. Since our equation already has a negative sign in front of 'h', when we plug in -1 we get |x-(-1)|
    2. Subtracting a negative is the same as adding a positive, so |x-(-1)| becomes |x+1|
    3. The vertical bars around 'x+1' are absolute values
    1. We're going to pick the same x-values we picked before: -2, -1, 0, 1, and 2
    2. Then we'll make a table and plug those x-values into the equation y = |x+1|
    3. Then again we can use those y-values to make ordered pairs we can plot to graph the equation
    4. Remember, the absolute value of a negative number is just that same number as a positive
    5. So |-2+1|, or |-1|, is just equal to 1
    1. Here again we have a horizontal translation, like we did before
    2. But this time, when we chose a negative value for 'h', our graph shifted LEFT instead
    3. Remember, even though we have a + sign in our equation, the value we originally chose for 'h' was negative: -1
    1. 'h' determines how far we slide the graph to the left or right
    2. Making a horizontal shift like this affects the x-coordinates of the graph
    3. For our parent function, when 'h' was 0, the vertex was at (0,0)
    4. When we picked 'h' to be 1, the new vertex became (1,0)
    5. When we picked 'h' to be -1, the new vertex became (-1,0)
    6. So 'h' tells us how far the x-coordinate moves from the original position of the parent function
    1. An absolute value equation with a non-zero 'h' will have a horizontal translation
    2. Remember, a horizontal translation is a shift to the left or right
    1. Be careful to account for the minus sign in the equation!
    2. Remember, the equation has the form y = |x-h|
    3. So if 'h' is positive, it will have a minus sign in front of it in the equation
    4. And if 'h' is negative, it will have a plus sign in front of it in the equation