Question
6.
Describe how the graph of y=|x| – 12 is like the graph of y=|x| and how it is different.
Describe how the graph of y=|x| – 12 is like the graph of y=|x| and how it is different.
Answers
GPT-4o mini
The graphs of \( y = |x| - 12 \) and \( y = |x| \) are related, but they have some key differences due to the transformation applied to the basic absolute value function.
### Similarities:
1. **Shape**: Both graphs have the same V-shape characteristic of absolute value functions. As \( x \) increases or decreases from the vertex (the point where the graph changes direction), both graphs rise at a constant rate.
2. **Vertex Structure**: Both graphs have a vertex, which is the lowest point on the graph of \( y = |x| \) and the lowest point of the shifted graph \( y = |x| - 12 \).
### Differences:
1. **Vertical Shift**: The primary difference is that the graph of \( y = |x| - 12 \) is shifted downward by 12 units compared to the graph of \( y = |x| \). The vertex of \( y = |x| \) is at the origin (0,0), while the vertex of \( y = |x| - 12 \) is at the point (0, -12).
2. **Y-Intercept**: The y-intercept of \( y = |x| \) is at (0,0), whereas the y-intercept of \( y = |x| - 12 \) is at (0, -12). This means that the entire graph is moved down, affecting all points along the y-axis.
3. **Range**: The range of \( y = |x| \) is \( [0, \infty) \), since the function outputs are non-negative. In contrast, the range of \( y = |x| - 12 \) is \( [-12, \infty) \), as the outputs are shifted downward and can now take values starting from -12 and going upwards.
### Summary:
In conclusion, while both graphs share the same V-shaped structure, \( y = |x| - 12 \) is a vertical downward shift of the basic absolute value graph \( y = |x| \) by 12 units, resulting in different vertices and y-intercepts.
### Similarities:
1. **Shape**: Both graphs have the same V-shape characteristic of absolute value functions. As \( x \) increases or decreases from the vertex (the point where the graph changes direction), both graphs rise at a constant rate.
2. **Vertex Structure**: Both graphs have a vertex, which is the lowest point on the graph of \( y = |x| \) and the lowest point of the shifted graph \( y = |x| - 12 \).
### Differences:
1. **Vertical Shift**: The primary difference is that the graph of \( y = |x| - 12 \) is shifted downward by 12 units compared to the graph of \( y = |x| \). The vertex of \( y = |x| \) is at the origin (0,0), while the vertex of \( y = |x| - 12 \) is at the point (0, -12).
2. **Y-Intercept**: The y-intercept of \( y = |x| \) is at (0,0), whereas the y-intercept of \( y = |x| - 12 \) is at (0, -12). This means that the entire graph is moved down, affecting all points along the y-axis.
3. **Range**: The range of \( y = |x| \) is \( [0, \infty) \), since the function outputs are non-negative. In contrast, the range of \( y = |x| - 12 \) is \( [-12, \infty) \), as the outputs are shifted downward and can now take values starting from -12 and going upwards.
### Summary:
In conclusion, while both graphs share the same V-shaped structure, \( y = |x| - 12 \) is a vertical downward shift of the basic absolute value graph \( y = |x| \) by 12 units, resulting in different vertices and y-intercepts.