College Algebra: Graph Quadratics Using Patterns

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Now I want to pull all these ideas together and show you how you can really graph complicated functions using the techniques that we've just developed. Let's take a look at this function right here. I want to graph f(x) = -3(x - 1)^2 + 2. It looks really, really threatening. The secret is to slowly unpeel it and do the different steps and see what this thing really is. What's at the very essence of this function? The very essence of this function is x^2. So let's start off with an x^2. So there's a parabola. What do we do next? Well, the next thing I notice is, "Okay, wait a minute. Now I've got that -3 in front of an x^2." Forget about the -1 right now. What does the -3 do? Well, a negative sign, remember, flips over the x-axis. It takes the happy face parabola and makes a sad face parabola, so, in fact, that negative sign does this.
Now, what does the 3 do? Well, the coefficient of 3 in front tightens things, it makes things sharper. So these wings come together, right? At 2, for example with x^2 we be at usually here with -x^2 we'd be at -4, now we'd be at -12. So that point is way down here, so it squeezes this together. So that 3 factor is a squeeze factor. It makes it tighter because the number is 3.
So that takes care of the -3 x^2. Now, what does at -1 do when I have an x - 1? Well, that's a shift in the x, which means I'm going to shift either this way or that way. Remember the classic mistake number 8. It's the shifting mistake. Remember the phrase--add to y, go high; add to x, go west. So here I'm subtracting, so I don't go west. I must go east by one unit, and then what does the +2 do? That's an increase in y, so I go high--1, 2, and there's the graph. So really neat. If you take your time with this you can actually graph a very complicated function pretty accurately.
Let me recap how I did that. I started with a standard parabola in general position. Why did I pick the parabola? Because I saw the very essence of this thing is just x^2. Then before I dealt with these shifting things I took a look at the coefficients here, and I saw -3(x^2). The minus sign--let me put it this way--the 3 tightened me up. Right? It made me steeper. The minus sign flipped me over the x-axis, so I've got that. The x - 1 term is a shift, and remember, add to x, go west, so in fact, this minus sign pushes me this way one unit, but then I take everything and increases by two--1, 2. So, in fact, there`s the graph of that. You can graph these really complicated functions. The key always is to find the essence of what's there, look at any multiples and get the flipping right, and then look at the shifting, at the very end, shift away.
All right. See if you can graph these much more complicated functions by first isolating an easy function and then transforming it into your exotic one. Good luck, and have fun.
Relations and Functions
Manipulating Graphs- Shifts and Stretches
Graphing Quadratics Using Patterns Page [1 of 1]