In , we've added to the outside of the function, so all of the outputs are increased by . Remember that on a graph, the outputs are represented by the -values, so this would increase all of the -values by . So, the graph of would be the same as the graph of , but moved up by .
We want to figure out which of the following is the graph of . In each, the gray dotted graph is the orignal graph of so you can more easily see how it changed.
Since we are looking for , we are looking for the graph to move down by 2, which is graph C. Graph A shifted up instead of down, which would be . Graph B changed the shape of the graph by stretching it, which you'll learn about in Section 9.3. Graph D shifted the graph to the side, which you'll learn about in Section 9.2.
To draw , we see that we are shifting the whole graph down 4. You may find it helpful to move the corner points down first, then connect them together to get the whole graph. So, you would move the point on the original graph down to . Similarly, you would move the point on the original graph down to and you would move the point on the original graph down to . From there, you can connect the lines to get the final graph, shown in blue below, where the gray dotted line is the orignal graph of .
To draw , we see that we are shifting the whole graph up 1. You may find it helpful to move the corner points up first, then connect them together to get the whole graph. So, you would move the point on the original graph up to . Similarly, you would move the point on the original graph up to and you would move the point on the original graph up to . From there, you can connect the lines to get the final graph, shown in blue below, where the gray dotted line is the orignal graph of .
Suppose the solid blue graph below is the graph of and the dotted red graph is the graph of :
We want to write a formula for in terms of . In other words, we start with and figure out how the graph changed to get to , then write how that changes the formula. Since the question asks for the formula "in terms of ", that means we will write somewhere in our answer. Looking at the graph, we see that is the graph of shifted up by . Therefore, that means we are adding to the outside of the function . So, our final answer is .
We want to write a formula for in terms of . In other words, we start with and figure out how the graph changed to get to , then write how that changes the formula. Since the question asks for the formula "in terms of ", that means we will write somewhere in our answer. Looking at the graph, we see that is the graph of shifted down by . Therefore, that means we are subtacting from the outside of the function . So, our final answer is .
Suppose and is the same as but shifted up . Let's write a formula for . Since we have an explicit formula for in this case, we can write an explicit formula for , too. Since we are shifting up by , that means we are adding to the outside of the function. So,
Since we have an explicit formula for in this case, we can write an explicit formula for , too. Since we are shifting down by , that means we are subtracting from the outside of the function. So,
Suppose and . Let's figure out what transformations took to . Comparing the two formulas, we see that only has added at the end, but has added at the end. That means that to get from to , we must have subtracted to the outside so that . That means there was a shift down by .
Comparing the two formulas, we see that only has subtracted at the end, but has added at the end. That means that to get from to , we must have subtracted to the outside so that . That means there was a shift down by .
