What Does the Constant 'k' do in y = |x|+k?

Summary

The bars around the 'x' are absolute value signs
'k' is a constant
The graph of y = |x| is a V shape with its vertex at the origin
y = |x| is the parent function of y = |x|+k
The absolute value of a negative number is just the positive version of that number
When 'k' is positive, the graph shifts upwards
When 'k' is negative, the graph shifts downwards

Notes

1. The bars around the 'x' are absolute value signs
2. y = |x| is the parent function of y = |x|+k
1. The graph of y = |x| is a V shape with its vertex at the origin
2. The bars around the 'x' are absolute value signs
3. y = |x| is the parent function of y = |x|+k
1. Let's set the constant 'k' equal to 1
2. y = |x|+k becomes y = |x|+1
3. The bars around the 'x' are absolute value signs
1. Let's pick -2, -1, 0, 1, 2 for our x-values
2. Plug each value for 'x' into y = |x| + 1
3. The absolute value of a negative number is just that number as a positive
4. So |-2|+1 becomes 2+1, or 3
5. |-1|+1 becomes 1+1, or 2
6. Non-negative numbers stay the same when you take their absolute value, so 0, 1, and 2 don't change
1. The ordered pairs found in the table are:
2. (-2,3), (-1,2), (0,1), (1,2) and (2,3)
3. Notice that the vertex of the graph shifted from (0,0) to (0,1)
4. We call this upward shift a vertical translation
1. Now let's set the constant 'k' equal to -1
2. y = |x|+k becomes y = |x|-1
3. The bars around the 'x' are absolute value signs
1. We'll create another table using the same x-values we used for the first graph
2. Again, taking the absolute value of negative numbers just makes them positive
3. So |-2|-1 becomes 2-1, or 1
4. And |-1|-1 becomes 1-1, or 0
5. Since 0, 1, and 2 are not negative, they stay the same when we take their absolute value
6. The bars around the 'x' are absolute value signs
1. The ordered pairs found in the table are:
2. (-2,1), (-1,0), (0,-1), (1,0) and (2,1)
3. Notice that the vertex of the graph shifted from (0,0) to (0,-1)
4. This downward shift is another type of vertical translation
1. When 'k' was positive 1 it moved the graph up one unit
2. When 'k' was negative 1 it moved the graph down one unit
3. The size and shape of the graph didn't change, just its vertical location
4. We call shifts like this vertical translations
1. The bars around the 'x' are absolute value signs
1. A vertical translation is a movement up or down on the coordinate plane
2. For example, when k = 1, the graph is moved up one unit
3. When k = -1, the graph is moved down one unit

What does the constant 'k' do in y = |x| + k?

Summary

Notes