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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 (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 (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)
.

1 answer

To determine the correct response, we need to analyze the translation applied to the function \( f(x) \).

Given that the point (1, 1) is on the original function \( y = f(x) \), if there is a horizontal translation, we need to know the direction and amount of translation.

  1. If the function is translated to the right by 2 units, the new point would be \( (1 + 2, 1) = (3, 1) \), and the translated function would be \( y = f(x - 2) \).

  2. If the function is translated to the left by 2 units, the new point would be \( (1 - 2, 1) = (-1, 1) \), and the translated function would be \( y = f(x + 2) \).

Since the question specifies a horizontal translation and the options indicate a rightward translation (\( f(x - 2) \)), the correct choice must reflect that the point (1, 1) moves to (3, 1) for this translation.

Thus, the response is:

  • 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 .

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