If an absolute value quantity is being multiplied by a factor greater than 1, then graph will be "skinnier" -- the opening of the v-shape will be more narrow. If an absolute value quantity is being multiplied by a factor that is less then one but greater than 0, then the opening of the v-shaped graph will be wider. If an absolute value quantity is being multiplied by a negative value, then graph will be flipped and open downwards.
These problems about absolute values don't use horizontal and vertical shifts because there's no adding or subtracting going on, instead they use what are called dilations. Dilations meaning my V is going to get either wider or skinnier. First let's talk about what this means.
If I were to take my input values whatever I wanted for x and then looked at their corresponding y output values and multiplied them by two, that means that my V-shape is going to get steeper more quickly or it's going to have a more steep slope.
What that looks like on the graph is that instead of having just the plain old V-shape for my absolute value which looks like this, now it's going to be twice as skinny. Or if you're using graph paper instead of going up one over, one up one over one to get your absolute value of V, now I'm going to go up two over one. Again this is just a really rough sketch because directions say sketch it, it doesn't have to be perfect and I'll tell you mine is not going to be perfect. But it's going to look like a V-shape that's twice as skinny as it was in the parent graph something like that. Let me erase that so you can see a little better.
That's what the graph of the absolute value of x multiplied by two would look like. What if I had a negative number outside? Negative means after you absolute value-ize your x quantities you would stick a negative in front of them meaning my range or my output would be all negative values. Usually absolute value looks like this. There's my parent graph only instead of having it be arranged in the positive y quadrants, I'm going to make my range now be negative. That's what it looks like, it's the same V equally as steep in this case only flipped into the negative quadrants for the y values.
Let's do part c. Part c is a little bit of combinations of both. I'm going to have a negative sign which means it's going to be an upside down V and then this one half means my V-shape is actually going to get wider.
Let me show you what I mean, if this is my parent graph, my slope when I drew those lines on graph paper would look like up one over, one up one over one. My slope here would look like up one over two, or down one over two. My slope is one over two, so on my sketch I'm going to go down one over two. Also notice I'm going down now because of that negative sign. Again all of my output values are going to be in the negative y range, something like that. I have a wide absolute value v. Keep in mind these parent functions and how they change according to these dilations and flips and you can do these graphs really, really quickly with pretty good precision.
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