To understand how the absolute value functions behave and to visualize them, let's describe each of the options and their graphs.
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\( y = |x - 3| \):
- This function has a vertex at \( (3, 0) \).
- The graph will have a V-shape opening upwards and will intersect the x-axis at \( x = 3 \).
- As \( x \) moves away from 3, \( y \) increases.
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\( y = |x + 3| \):
- This function has a vertex at \( (-3, 0) \).
- The graph will also have a V-shape opening upwards and will intersect the x-axis at \( x = -3 \).
- As \( x \) moves away from -3, \( y \) increases.
-
\( y = |x| + 3 \):
- This function has a vertex at \( (0, 3) \).
- The graph will have a V-shape opening upwards and will be shifted up by 3 units compared to the standard absolute value function.
- It intersects the y-axis at \( (0, 3) \) and at \( y = 3 \) for both \( x = -3 \) and \( x = 3 \).
-
\( y = |x| - 3 \):
- This function has a vertex at \( (0, -3) \).
- The graph will have a V-shape opening upwards, but it is shifted down 3 units compared to the standard absolute value function.
- It intersects the y-axis at \( (0, -3) \).
Summary of the Graphs
- \( y = |x - 3| \): V-shape, vertex at (3, 0).
- \( y = |x + 3| \): V-shape, vertex at (-3, 0).
- \( y = |x| + 3 \): V-shape, vertex at (0, 3); entire graph is shifted up.
- \( y = |x| - 3 \): V-shape, vertex at (0, -3); entire graph is shifted down.
Upper Function:
To determine which function creates the "upper" function in the graph, it depends on how we interpret "upper". If you mean the function that has the uppermost vertex position relative to the others, then:
- The highest vertex would clearly be from \( y = |x| + 3 \) because its vertex at (0, 3) is higher than the others:
- \( |x - 3| \) and \( |x + 3| \) have vertices at (3, 0) and (-3, 0) respectively, while \( |x| - 3 \) has its vertex at (0, -3) which is the lowest.
Thus, the function that creates the upper function in the graph is \( y = |x| + 3 \).
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