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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 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 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.

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

1 answer

To determine which graph shows that \( y = f(x - 3) \) is the result of the function \( y = f(x) \) being translated by 3 units, we need to understand how horizontal translations work.

The function \( y = f(x - 3) \) represents a shift of the original function \( y = f(x) \) to the right by 3 units. This means that for every point (x, y) on the original graph, the corresponding point on the translated graph will now appear at (x + 3, y).

In summary, to identify the correct graph:

  1. Look for a transformation that moves all points of the original graph to the right by 3 units.
  2. The translated graph should appear to be exactly the same shape and structure as the original graph, but shifted along the x-axis.

Without seeing the graphs mentioned in the choices, I’m unable to point out which one illustrates this correctly. However, examine each of the graphs and see if you can identify one where the solid line (translated graph) corresponds to the dotted line (original graph) shifted to the right by exactly 3 units. This will be the correct graph representing \( y = f(x - 3) \).