The equation of the graph of the absolute value function that is shifted 3 units to the right, shifted 5 units up, and is reflected over the x-axis is i(x) = - |x - 4| + 3.
What is function?
A functional is a real-valued, often function-based, function on a vector space. For instance, any differentiable function is given a number by the energy functional on the unit disk. However, for the functional to be continuous, the vector space of functions must have the proper topology.
Refer to attached graph
Parent function:
f(x) = |x|, solid black on the graph
Transformations
1. Reflection over x-axis: f(x) → -f(x)
g(x) = -|x|, dotted blue on the graph
2. Horizontal shift 4 units to the right: g(x) → g(x - 4)
h(x) = -|x - 4|, dotted green on the graph
3. Vertical shift 3 units up: h(x) → h(x) + 3
i(x) = - |x - 4| + 3, solid red on the graph
This is the final function i(x) = - |x - 4| + 3.
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