To translate the function \( f(x) = x^2 + 22x + 58 \) 4 units to the right and 16 units up, we can follow the steps below:
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Translation to the Right: Translating a function to the right by \( h \) units means replacing \( x \) with \( x - h \). Here, \( h = 4 \). Thus, we replace \( x \) with \( x - 4 \):
\[ f(x - 4) = (x - 4)^2 + 22(x - 4) + 58 \]
Now, expand this:
\[ = (x - 4)^2 + 22x - 88 + 58 \] \[ = (x - 4)^2 + 22x - 30 \]
Now, we need to expand \( (x - 4)^2 \):
\[ (x - 4)^2 = x^2 - 8x + 16 \]
So, substituting that back, we get:
\[ f(x - 4) = x^2 - 8x + 16 + 22x - 30 \] \[ = x^2 + 14x - 14 \]
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Translation Upwards: Translating a function upward by \( k \) units means adding \( k \) to the function. In this case, \( k = 16 \):
\[ f(x - 4) + 16 = (x^2 + 14x - 14) + 16 \] \[ = x^2 + 14x + 2 \]
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Converting to Vertex Form: To write the new function in vertex form, we need to complete the square for \( x^2 + 14x + 2 \).
\[ = (x^2 + 14x + 49) - 49 + 2 \] \[ = (x + 7)^2 - 47 \]
Thus, the vertex form of the new function after the required translations is:
\[ (x + 7)^2 - 47 \]
Therefore, the correct option from the given choices is:
\[ \boxed{(x + 7)^2 - 47} \]