Question
12.
Describe how the graph of y=|x| – 4 is like the graph of y=|x| and how it is different.
Describe how the graph of y=|x| – 4 is like the graph of y=|x| and how it is different.
Answers
GPT-4o mini
The graph of the function \( y = |x| - 4 \) shares similarities and differences with the graph of \( y = |x| \).
### Similarities:
1. **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.
2. **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:
1. **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).
2. **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).
3. **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.
### Similarities:
1. **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.
2. **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:
1. **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).
2. **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).
3. **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.