Use the image to answer the question.

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 (1,1)
, 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 plus 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 (1,3)
, and the translated function is y=f(x−2)
.

1 answer

To determine the location of the point after a horizontal translation of the original function \( y = f(x) \) that goes through the point (1, 1), we need to consider the nature of horizontal translations.

  • A horizontal translation to the right by \( h \) units transforms \( f(x) \) into \( f(x - h) \).
  • A horizontal translation to the left by \( h \) units transforms \( f(x) \) into \( f(x + h) \).

In your question, it appears that the point (1, 1) initially lies on the curve defined by \( y = f(x) \). Without knowing specifically how many units the translation has occurred (to the left or right), we apply the translation results to determine the new point and function.

  1. If translated to the right (for example \( h = 2 \)):

    • Original point: \( (1, 1) \)
    • Translated point: \( (1 + 2, 1) = (3, 1) \)
    • Translated function: \( f(x - 2) \)
  2. If translated to the left (for example \( h = 2 \)):

    • Original point: \( (1, 1) \)
    • Translated point: \( (1 - 2, 1) = (-1, 1) \)
    • Translated function: \( f(x + 2) \)

Given the responses you provided, it seems you're looking for translations specifically either to the right or left, likely by 2 units based on the options showing 2 as the distance.

The correct match from your responses, through reasoning, is that the point ends up at \( (3, 1) \) if translated to the right by 2 units, leading us to choose:

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

This is repeated in both last options. Therefore, either of these statements corresponds to the condition of having the point after translation, leading us to conclude:

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

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