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

1.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.

2.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.

3.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.

4.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.

GPT-4o mini · Answer

To determine which graph shows that $ y = f(x - 3) $ is a translation of $ y = f(x) $ by 3 units to the right, we need to consider how the translation affects the graph. When you translate a function $ y = f(x) $ horizontally to the right by 3 units, the new function, $ y = f(x - 3) $, will have all of its points shifted 3 units to the right. This means that if a point $ (a, f(a)) $ is on the original graph $ y = f(x) $, then the point $ (a + 3, f(a)) $ will be on the translated graph $ y = f(x - 3) $. Now, let’s analyze the options: 1. In the first option, if both curves are translated and you visually check where the original function is positioned, it must be compared to where the translated function has moved. 2. The second option with the parabolas may show a shift, but we need to check specifically for a shift of 3 units to the right. 3. The third option shows lines, but we should see if they align appropriately or just overlap. 4. The fourth option also shows lines, which might correspond similarly. You would be looking for something that distinctly moves every point of the original graph 3 units to the right without changing its vertical position. The solid line graph should begin at the original x-value plus 3 for corresponding y-values. Without the ability to see the actual graphs, it's essential to focus on the descriptions. Generally, the graph where the original function and the translated function maintain their form but shifted rightwards indicates that $ y = f(x - 3) $ is the true representation of a 3-unit right translation. If you are still uncertain, I would suggest picking the graph with the translation that seems most clear in moving the points distinctly to the right.