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Two curves drawn on coordinate plane. Both axes range from negative 5 to 5 in unit increment.

The original function y=f(x)
in the graph (solid line) goes through the point (1,1) and is translated using a horizontal translation. Where is this point located after the translation, and what is the translated function?

(1 point)
Responses

The point is located at (1,3), and the translated function is y=f(x−2).
The point is located at left parenthesis 1 comma 3 right parenthesis , and the translated function is f left parenthesis x minus 2 right parenthesis .

The point is located at (1,1), and the translated function is y=f(x−2).
The point is located at left parenthesis 1 comma 1 right parenthesis , and the translated function is f left parenthesis x minus 2 right parenthesis .

The point is located at (3,1), and the translated function is y=f(x−2).
The point is located at left parenthesis 3 comma 1 right parenthesis , and the translated function is f left parenthesis x minus 2 right parenthesis .

The point is located at (3,1), and the translated function is y=f(x+2).
The point is located at left parenthesis 3 comma 1 right parenthesis , and the translated function is f left parenthesis x plus 2 right parenthesis .
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1 answer

To determine the translated point and function, we need to establish how the original function \( y = f(x) \) is being translated.

Given that the original point is \( (1, 1) \) and we assume the translation is horizontal (which usually indicates a shift along the x-axis), we denote translations to the right as \( f(x - h) \) and translations to the left as \( f(x + h) \), where \( h \) is the amount of translation.

  • If the function is translated to the right by 2 units (which would mean \( h = 2 \)), the new function would be \( y = f(x - 2) \).
  • If the function is translated to the left by 2 units, the new function would be \( y = f(x + 2) \).

For the case of a right translation \( (h = 2) \):

  • The translated point would be \( (1 + 2, 1) = (3, 1) \).

For the case of a left translation \( (h = -2) \):

  • The translated point would be \( (1 - 2, 1) = (-1, 1) \).

Without additional visuals, it seems the most likely scenario is a translation to the right, resulting in the point moving from \( (1, 1) \) to \( (3, 1) \) with the function given by \( y = f(x - 2) \).

Thus, the correct response would be:

The point is located at (3,1), and the translated function is y=f(x−2).

Translated in a clearer format:

The point is located at (3, 1), and the translated function is \( y = f(x - 2) \).