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

is the result of the function y=f(x)
being translated by 3 units?(1 point)
Responses

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 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. 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.

Answer 1

To find which graph represents 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 following characteristics:

When a function is translated horizontally, the graph of the new function \( y = f(x - h) \) will shift to the right by \( h \) units if \( h \) is positive. Thus, \( y = f(x - 3) \) indicates that the graph has been shifted 3 units to the right.

Now, let's check each of the described options:

1. Two parabolas - If the original dotted line (representing \( y = f(x) \)) is located at a certain position, and the solid line (representing \( y = f(x - 3) \)) is shifted right by 3 units, this would show the desired translation.

2. Two curves - Similar reasoning applies. If the solid curve is a rightwards shift of the dotted curve by 3 units, then this is the correct representation.

3. Two lines (first set) - For this comparison, if the solid line is positioned 3 units to the right of the dotted line, it confirms that the function has been translated correctly.

4. Two lines (second set) - Again, we would look for a similar shift.

To identify the correct choice purely from the description, we would look for any graph where the solid line appears distinctly 3 units to the right of the dotted line.

Without seeing the specific visuals, you should choose the graph that clearly displays the solid graph shifted 3 units to the right of the dotted graph to indicate that \( y = f(x - 3) \) is a translation of \( y = f(x) \).