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Shifts in Absolute Value Graphs - Problem 1
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If a value is being added or subtracted to the absolute value quantity (meaning it is outside of the absolute value signs), then the graph will shift up (if the value is being added) or down (if the value is being subtracted). If a value is being added or subtracted to the x variable inside the absolute value signs, then the graph will shift to the left (if the value is being added) or to the right (if the value is being subtracted).

When you're asked to make graphs you could always make a table of values but it can be a drag. I like using short-cuts especially when the directions tell you to sketch. That means it doesn't have to be totally precise.

Okay so this tells me I'm going to take my absolute value v-shape and move it up 2 on the y axis. Here's what I mean, usually absolute value graphs have a v where the bottom is at the origin. Also these lines like the sides of the v go up through the exact corners of your boxes or what we call a 45 degree angle.

What I'm going to do is take that graph, I'm going to make it dashy so that's not my real answer, my real answer is going to be that exact shape moved up 2 on the y axis. There it is it should be equally as steep and it's the same shape just moved up 2. That's it I didn't have to make a table or anything.

What I'm doing in my head, like the reason why these shifts work, is that I'm taking my absolute value quantities and then my output numbers for every single one is being added to two. That means output or y gets shifted up two. That's why this works, b is really similar.

For b I'm going to take my absolute value parent shape which would be a v like this only now I'm going to move it down two on the y axis. It's going to be a v again that's equally as steep as my parent function v should be 45 degree angles if you know what that means. If you're drawing on graph paper it's going to go exactly through the corners of every box. That's it, that's my answer for part b, the absolute value of x take away 2.

These numbers outside the absolute value signs represent vertical shifts. I'm going to erase this and then do part c and d which are horizontal shifts.

Part c I have a plus two inside my absolute value. Usually positive means move to the right, right? But when you have a number inside the argument or inside the absolute value tracks, that means move to the left, if it's positive. So I'm going to have my parent graph which is usually here with the center at the origin, oops that's pretty ugly it should be 45 degree here like that. Now I'm going to move that shape left two, left two like that. That's what my answer for graph c would look like.

Graph d is really similar only now I'm going to move to the right instead of being at the origin I move two units to the right make my v-shape like that.

This is why I really like shifting rules, if you can memorize these rules and how to work with them, these problems go by so, so quickly and you don't have to make those tables that take all day.

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