Shifting Graphs

Exercises 27...

Question

Shifting Graphs

Exercises 27–36 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation.

y = x³ Left 1, down 1

Accepted Answer

Hello. In this video, we are going to be writing the new equation after shifting the given graph as stated below. We are also going to plot both of the graphs on the same coordinate axis. So, the graph given to us is Y is equal to 27 X cubed, and we want to shift this equation 3 units to the left and 3 units down. Let's go ahead and start this problem by graphing Y is equal to 27 X cubed. In order to graph this problem, let's just go ahead and plug in some values for X. We are going to plug in -1, 0, and 1. Now, when we plug in the value of -1 into equal to 27 X cubed, we get the value of -27. When we plug in 0, we get 0, and when we plug in 1, we get 27. So, this gives us the 3 points to plot for Y is equal to 27 X cubed. We have 1 point located at the origin, we have 1 point located located at 1.27. And we have 1 point located at -127. Now, what about the shape of the graph? Well, since the order of this graph is 3, we have a cubic graph, and a cubic graph is going to have a snake-like shape, or it's gonna look like a backwards S. So this is going to be the graph of Y is equal to 27 X cubed. Now that we've graphed Y is equal to 27 X cubed, let's go ahead and apply the transformations. So we're going to start with the phrase, left 3 units. Now, when we are trying to shift this graph left 3 units, we are talking about a horizontal shift. A horizontal shift is usually indicated by an X value, such as X minus H. But since we are shifting the graph 3 units to the left, our horizontal shift is going to have the form X minus negative H, or in other words, it'll be X plus H. Now, since we are shifting to the left 3 units, H is going to have the value of 3. So we are going to replace the value of X in the given equation to be X + 3. That will allow us to rewrite the equation to be Y is equal to 27 multiplied by the quantity X + 3, raised to the power of 3. Now let's go ahead and apply the transformation down 3 units. Now, when we are trying to shift the graph down 3 units, what this means is that we want to subtract every value of the graph by 3. In order to apply this vertical shift, all we have to do is just subtract 3 from the entire equation, and that will allow us to rewrite the equation to be Y is equal to 27 multiplied by X + 3 cubed, then minus 3, and that's how we apply the shift down 3 units. This is going to be the equation of the graph after applying the given transformations. And now let's go ahead and plot the given graph. Now, since we are shifting the whole graph left 3 units, we are going to move every X value in our original plot to the left 3 units. Or in other words, we are going to be subtracting 3 from every graph. And so what this means is that for the 0.1.27, the X value is now going to be -2. And for down 3 units, we have to just subtract 3 from all the Y values. So we will end up with 3 points in total. The first point is going to be at -2.24. The second point will be at -3-3, and the final point will be located at -4-30. If we plot these points, we will first start by plotting -2.24. That is going to be located roughly here. This will be -2024. Now, we also have the origin 00 being shifted to the left 3 and down 3. So that's going to be located at -3 -3. And finally, we have the 0.-4 -30, which is going to be the very bottom of this graph. Now again, because this is still a cubic graph, our shifted graph is going to have the same shape as our original graph, so we will still have the S-like structure. And this is going to be the shape of the graph after shifting it to the left 3 units and down 3 units. So, I hope this video helps you in understanding how to approach this problem, and I will go ahead and see you all in the next video.

Shifting Graphs

Exercises 27–36 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation.

y = x³ Left 1, down 1

Key Concepts

Graph Shifting

Horizontal Shift

Vertical Shift

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