We start with the original function modeling the height of the diver above the water, which is given by:
\[ h(x) = -(x - 2)^2 + 9 \]
This function describes a parabolic path where the maximum height of the diver (at \( x = 2 \)) is 9 feet above the water.
a. The diving board is lowered by 3 feet
If the diving board is lowered by 3 feet, the maximum height above water will also decrease by 3 feet. The new function will be:
\[ h(x) = -(x - 2)^2 + (9 - 3) = -(x - 2)^2 + 6 \]
b. The diving board is shortened by 1 foot
If the diving board is shortened by 1 foot, this means that the new maximum height is reduced by 1 foot, but it is still the height above the water. Thus, the new function will be:
\[ h(x) = -(x - 2)^2 + (9 - 1) = -(x - 2)^2 + 8 \]
c. The diving board is raised by 2 feet
If the diving board is raised by 2 feet, the maximum height now increases by 2 feet. The new function will be:
\[ h(x) = -(x - 2)^2 + (9 + 2) = -(x - 2)^2 + 11 \]
d. The diving board is extended by 0.5 feet
If the diving board is extended horizontally (but not changed in height), it does not affect the height of the diver at any given time. The function remains the same:
\[ h(x) = -(x - 2)^2 + 9 \]
However, if we assume that this change affects the time taken for the jump (for example, shifting the jump left or right), the vertex may need to be adjusted accordingly. Since the problem doesn't specify a precise effect on timing, we'll keep the original function for simplicity.
e. Is it possible for h(x) to undergo a reflection over the x-axis based on the context of the problem?
A reflection over the x-axis would change the function to:
\[ h(x) = (x - 2)^2 - 9 \]
In terms of context, this would mean that instead of measuring height above water, you would be measuring depth below water. Since this is a modeling of a diver jumping, it does not make sense for the function to represent depths below the water after jumping since we are interested in heights above the water. Thus, it is not contextually appropriate for \( h(x) \) to reflect over the x-axis. The reflection indicates a fundamentally different scenario than the one being modeled (i.e., a diver would not typically be measured below the water when jumping into it).
In conclusion, the reflection over the x-axis is not possible within the context of the diver jumping above the water surface.
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