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Shifts in Absolute Value Graphs - Problem 3
Instead of using a table of values, use shifts and dilations to sketch a graph. Remember the following rules when sketching a graph: 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). 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.
If I was an Algebra One student looking at this problem on my homework I'd be pretty intimidated. To be honest with you I'm Math teacher and looking at this problem I'm a little intimidated. It would be a huge, huge time eater-upper if I had to go through and make a table of values. If I had to choose Xs, plug them in one by one and find the y values, put the dots on the graphs and sketch it. That's why I really like using shifts and dilations, when it comes to absolute value graphs.
I know the parent graph looks like a v and in all of these numbers just affect how skinny or wide it is and also where my vertex of the v goes. Let's talk it through one by one.
This 2 being multiplied the x means that my graph is going to be wide, no it doesn't my fault it's going to be skinny. It's going to be half as skinny as it would if it didn't have that two there. Skinny v-shape Then this -3 inside the parentheses or inside the absolute value grouping usually tells me negative would move to the left but since it's inside I'm going to move three units, right. Three units right and then that +4 at the end means I'm going to move four units up from my vertex. One thing that sometimes confuses students is they want to do this distributing; they want to do 2 times -3 and think about six units to the right. You don't have to do that in fact please don't because when you distribute your messing with the absolute value process and technically messing up the order of operations. So let's just use these shifts and we'll have the graph very quickly. Okay so first thing I'm going to do is find where the bottom of my v will be and then I'll draw the skinny or wideness after that.
Move three units to the right and four units up. Okay here's my graph I'm going to move three to the right, four units up that's where the bottom of my v is going to be. From there I'm going to draw a skinny v-shape, usually to draw an absolute values v I would go up one over one, up one over one. Since I have two there I'm going to go up two over one like a slope, up two over one in both directions. Up two over one, up two over one looks something like that, up two over one running into my origin.
That's it, that's a pretty good graph and since the directions just told me to sketch, it doesn't have to be exact. It doesn't have to be precise which is a nice thing. If you have graph paper you could do this exactly by counting the slope over two up one after you've moved the bottom of your v shape. The cool I want to leave you guys with when it comes to these kinds of shifting rules, is that they don't only apply to absolute values. Don't feel like your investing time working on these rules only for this parent function. It's true for all kinds of functions and you'll start seeing more and more of them as you move through your subsequent Math classes.