Each of the function graphs represents an original function, marked by a dotted line graph, and a translated graph, marked by a solid line graph. Which graph shows that y=f(x−3)

Question

Each of the function graphs represents an original function, marked by a dotted line graph, and a translated graph, marked by a solid line graph. Which graph shows that  y=f(x−3)
  is the result of the function  y=f(x)
  being translated by 3 units?(1 point)
Responses

Two lines drawn on coordinate plane. X-axis range from negative 3 to 9 and y-axis range from negative 3 to 3 in unit increment.
Image with alt text: Two lines drawn on coordinate plane. X-axis range from negative 3 to 9 and y-axis range from negative 3 to 3 in unit increment.

Two curves drawn on coordinate plane. Both axes range from negative 5 to 5 in unit increment.
Image with alt text: Two curves drawn on coordinate plane. Both axes range from negative 5 to 5 in unit increment.

Two lines drawn on coordinate plane. Both axes range from negative 5 to 5 in unit increment.
Image with alt text: Two lines drawn on coordinate plane. Both axes range from negative 5 to 5 in unit increment.

Two parabolas drawn on coordinate plane. X-axis range from negative 9 to 4 and y-axis range from negative 1 to 9 in unit increment.
Image with alt text: Two parabolas drawn on coordinate plane. X-axis range from negative 9 to 4 and y-axis range from negative 1 to 9 in unit increment.
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GPT-4o mini · Answer

To identify which graph shows the function \( y = f(x - 3) \) as a translation of the original function \( y = f(x) \) by 3 units to the right, we need to look for the graph where the solid line (representing \( y = f(x - 3) \)) is shifted to the right compared to the dotted line (representing \( y = f(x) \)).

In the case of a horizontal translation to the right, every point on the graph of the original function \( y = f(x) \) will be moved 3 units to the right to create the graph of \( y = f(x - 3) \). This means:

If \( y = f(x) \) has a point at \( (a, b) \), then \( y = f(x - 3) \) will have a corresponding point at \( (a + 3, b) \).

From the descriptions given, the correct graph will show that the solid line is shifted to the right by 3 units compared to the dotted line.

While I cannot see the graph images, you should look for the one that most clearly shows the solid line (representing \( y = f(x - 3) \)) is the same shape as the dotted line (representing \( y = f(x) \)) but translated 3 units to the right. This will be the graph that visually represents that \( y = f(x - 3) \) is indeed the result of translating \( y = f(x) \) by 3 units.