The graph of the function \( y = |x| - 4 \) shares similarities and differences with the graph of \( y = |x| \).
Similarities:
- Shape: Both graphs have a V-shape because they are based on the absolute value function. This means they exhibit the same fundamental structure, where the left side of the V corresponds to negative x-values and the right side corresponds to positive x-values.
- Vertex: The vertex (the point at which the V shape turns) of both graphs is aligned vertically. In other words, both graphs have a point where the y-value is at a minimum.
Differences:
- Vertical Shift: The graph of \( y = |x| - 4 \) is vertically shifted downward by 4 units compared to the graph of \( y = |x| \). The vertex of \( y = |x| \) is at the origin (0,0), while the vertex of \( y = |x| - 4 \) is at the point (0,-4).
- Y-Intercept: The y-intercept of the graph of \( y = |x| \) is (0, 0), whereas the y-intercept of \( y = |x| - 4 \) is (0, -4).
- Output Values: For all x-values, \( y = |x| - 4 \) will output values that are 4 units less than those of \( y = |x| \) at the same x-value.
In summary, the graph of \( y = |x| - 4 \) is essentially the graph of \( y = |x| \) shifted down by 4 units, resulting in the same V-shape but positioned lower on the coordinate plane.
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