Video: Combined Transformations of the Graph of an Absolute Value Function

The following graph is a transformation of the graph of π¦ = |π₯|. What is the function it represents? Write your answer in a form related to the function.

02:28

Video Transcript

The following graph is a transformation of π¦ equals the absolute value of π₯. What is the function it represents? Write your answer in a form related to the function.

Here, weβre looking at the general π¦ equals the absolute value of π₯ with some transformations. π, first looking at π and what it would do to the absolute value graph. If it would be positive, the graph would look like a V; it would be facing upwards. If π were negative, it would be the exact opposite; it would be opening downwards. If the absolute value of π, meaning weβre not looking at the sign anymore, if the absolute value of π is greater than one, this will sh- vertically stretch the graph. If π is between zero and one, so if itβs a fraction, it will be a vertical compression of the graph.

Now looking at β, that will be the horizontal shift, so moving left and right. And finally, π will be the vertical shift, how much it moved up or down.

So first, looking at our graph, it is upside down. So that means π must be negative. And next, to determine what π actually is, so if itβs β the absolute value will be bigger than one or if it will be between zero and one, itβs essentially like the slope of the line. And we can see, if we go from the peak to the right, it goes down one, right one. So π is one, which this kind of makes sense. Looking at that, itβs negative one; itβs the slope.

Next, we look at β, the horizontal shift. So itβs π₯ minus the horizontal shift. So the original π¦ equals the absolute value of π₯ graph would start at zero. So-so if the original graph started at zero, zero, our new graph started one space to the left, which would be a negative one shift.

And then finally, how much would it have moved up? It went up four.

So to simplify this, we have π¦ equals negative one absolute value π₯ plus one plus four. We could use the commutative property of addition because weβre adding four to that absolute value. So we could put the four first and then put minus. We donβt have to put the one in front of the absolute value. So we could have π¦ equals four minus the absolute value of π₯ plus one.