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Graphing the Transformation y = f(x - h) - Problem 2
Let's graph another transformation. This time y equals ½the quantity x minus 4 cubed minus 5. Now what parent function is being transformed here? It's the cubing function; F(x) equals x³. Just to remind you, the graph of that function looks like this. It's got a little twist in it. It passes through 0,0 always increasing goes through 1,1 and -1,-1.
Now I can start off with those as key points, when I'm plotting my table of key points. Let's change the variable to u, because I'm going to do some transformations here. So -1,-1, 0,0, and 1,1. So those will be my key points.
Now I'm going to make the substitution u equals x minus 4. So if u equals x minus 4, then I add 4 to both sides. I get u plus 4 equals x. What this tells me is when I'm making my table of values, I need to add 4 to the u values to get my x values from my transformed table. This is going to be ½ x minus 4 cubed minus 5.
So let me do that. I add 4 to each of these values, and I get -1 plus 4, 3. 0 plus 4, 4. 1 plus 4, 5. Then this formula tells me that whatever happens to my u³, I have to multiply that by ½, and subtract 5. So multiply -1 by a ½ I get -½, and subtract 5, I get -5.5. Multiply by ½ and subtract 5, I get 5. Multiply by ½, I get +½ minus 5 is -4.5. These numbers are kind of close together, so I think I might plot a couple more points. Let's do say -2, and 2. -2³ is -8, 2³ is 8. So remember we have to add 4 to our u values. So -2 plus 4 is 2. 2 plus 4 is 8.
Then I take the y values, these u³ values multiplied by ½, and subtract 5. So -8 times ½ is -4, minus 5, -9. 8 times ½ is 4 minus 5 is -1. These are much better points. So let me focus on these three points. Also notice 0,0 which is this middle point on the graph, it's what we call an inflection point, where the graph has a twist in it. It goes from curving downward, to curving upward. That's at 0,0, but after the transformation, it's at 4,-5. This is really important. So the inflection point will be at 4,-5. Now I've scaled the x and y axis differently just it'll allow the graph to fit better. I have 4,-5, that's about here.
Now I'll plot these two points. I've got 2,-9. Here is 2, and that's -10 so -9 will be right here. Then I have 6,-1. 5, 6, -1 is here. I think I'm ready to graph this. Remember this is the inflection point, so it's going to curve down like this, and curve up like this. I'll just mark the coordinates. This is 4,-5. That's my graph for my transformed cubic.