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What is the transformation of the parent function f(x) = x^2 to f(x) = (x - 4)^2 - 5?

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Final answer:

The transformation from f(x) = x^2 to f(x) = (x - 4)^2 - 5 includes a horizontal shift of 4 units to the right and a vertical shift of 5 units down. The horizontal shift is caused by the (x - 4) inside the squared term, and the vertical shift is due to subtracting 5 from the entire function.

Step-by-step explanation:

The transformation of the parent function f(x) = x^2 to f(x) = (x - 4)^2 - 5 involves two main steps:

  • Horizontal shift
  • Vertical shift

A horizontal shift to the right of the graph occurs due to the substitution of x with (x - 4). This translates the function by 4 units to the right. Subsequently, a vertical shift downwards by 5 units is represented by the subtraction of 5 from the entire function, resulting in the final equation f(x) = (x - 4)^2 - 5. When contemplating how to undo the square in an equation such as a^2 = c^2 - b^2, one would take the square root of both sides to solve for a.

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