Shifts in Absolute Value Graphs - Concept
Once we learn about absolute value graphing, we can use shifts in absolute value graphs to more easily graph new absolute value equations. We can use specific rules for the addition of constants within the equation to shift the graph horizontally or vertically. These techniques can also be used when graphing parabolas.
When you're asked to make a graph in your Algebra class or any of your Math classes, you could always make a table of values where you chose x points substitute them in one by one and find their corresponding y values, but that takes all day, I don't like doing that I don't know about you guys. I like shortcuts, so when it comes to most graphs in Math class, you have what's called the parent function or a parent shape and from there you just shift it around the graph. You move it side to side, up and down sometimes you turn it up-side-down or make it skinny or wide.
This is how you know what to do, first thing to know is that all absolute value graphs looks like v's. We're talkin, excuse about v's with a sharp point at the bottom not the u parabola shape but v's. From there you have a whole bunch of shifts and here's how you know what to do. Shifting means you take the v and you move it either side to side or up and down. If you have an absolute value +a where the a is outside the absolute value, you move it up a units along the y axis. If it's absolute value of x take away a you move down. If you have a -a inside the absolute value you're going to move a units to the right. Now this is tricky because usually negative values mean move to the left, right so keep that straight in your head. That's one kind of trick to these I personally think, if you have a +a in there usually pluses to the right but here you are going to shift left, those are tricky.
The other thing you might do with your V is make it skinnier or wider based on what we call dilations, if absolute value of x is being multiplied by a, and if a is an integer or a fraction that's greater than 1 when your absolute value a it's going to become a skinnier v. If the absolute value of a is less than 1 but greater than zero yeah absolute value is always greater than zero it's going to be a wider absolute value of v. And then your whole v would turn up-side-down if a were a negative number.
And this will make a lot more sense once you start seeing problems and doing combinations of these and again you guys you can always make a table. But I'm going to tell you now these shifting rule show up not only in Algebra with absolute values but also with Parabolas and Quadratics. And then they'll show up again and again and again once you start doing fancier graphs in your advance Math classes. So if I were you and if I like shortcuts and I do, I would start memorizing these rules for shifts and dilations and flips.