Shifting of graphs: Consider the graphs below and answer the following questions.

Thinking of Graph B as f(x), which graph represents

 f(x-7): In the original graph (Curve B), the minimum occurs for x = 0. That is, when the argument of the function is 0, we have the minimum. So we must determine when the argument of f(x-7) will have an argument of 0. Since that will occur when x is 7, the graph we are looking for will have its minimum at x = 7. Will the minimum still have a y coordinate of 0? Yes, so we can eliminate curves G, I, A, C, D, and F. Looking at the remaining curves H and E, we see that curve E has its minimum in the correct place, x = 7, and represents f(x-7).

f(x+7): In the original graph (curve B), the minimum occurs for x = 0. That is, when the argument of the function is 0, we have the minimum. So we must determine when the argument of f(x+7) will have an argument of 0. Since that will occur when x is -7, the graph we are looking for will have its minimum at x = -7. Will the minimum still have a y coordinate of 0? Yes, so we can eliminate curves G, I, A, C, D, and F. Looking at the remaining curves H and E, we see that curve H has its minimum in the correct place, x = -7, and represents f(x+7).

f(x) + 3: In the original graph (curve B), the minimum occurs for x = 0. That is, when the argument of the function is 0, we have the minimum. So we must determine when the argument of f(x) + 3 will have an argument of 0. Since that will occur when x is 0, the graph we are looking for will have its minimum at x = 0. Will the minimum still have a y coordinate of 0? No, it will be at 0 + 3, so we can eliminate curves H, I, B, C, E, and F. Looking at the remaining curves with their minima at y = 3 (G, A, D), we see that curve A has its minimum in the correct place, x = 0, and represents f(x) + 3.

f(x) - 3: In the original graph (curve B), the minimum occurs for x = 0. That is, when the argument of the function is 0, we have the minimum. So we must determine when the argument of f(x) - 3 will have an argument of 0. Since that will occur when x is 0, the graph we are looking for will have its minimum at x = 0. Will the minimum still have a y coordinate of 0? No, it will be at 0 - 3, so we can eliminate curves G, H, A, B, D, and E. Looking at the remaining curves with their minima at y = -3 (I, C, F), we see that curve C has its minimum in the correct place, x = 0, and represents f(x) - 3.

f(x-7) + 3: In the original graph (curve B), the minimum occurs for x = 0. That is, when the argument of the function is 0, we have the minimum. So we must determine when the argument of f(x-7) + 3 will have an argument of 0. Since that will occur when x is 7, the graph we are looking for will have its minimum at x = 7. Will the minimum still have a y coordinate of 0? No, it will be at 0 + 3, so we can eliminate curves H, I, B, C, E, and F. Looking at the remaining curves with their minima at y = 3 (G, A, D), we see that curve D has its minimum in the correct place, x = 7, and represents f(x-7) + 3.

f(x-7) - 3: In the original graph (curve B), the minimum occurs for x = 0. That is, when the argument of the function is 0, we have the minimum. So we must determine when the argument of f(x-7) - 3 will have an argument of 0. Since that will occur when x is 7, the graph we are looking for will have its minimum at x = 7. Will the minimum still have a y coordinate of 0? No, it will be at 0 - 3, so we can eliminate curves G, H, A, B, D, and E. Looking at the remaining curves with their minima at y = -3 (I, C, F), we see that curve F has its minimum in the correct place, x = 7, and represents f(x-7) - 3.