How to transform graphs of functions?

So you've encountered something like:

g(x) = | 4 f( |x + 1| - 3 ) + 5 |

What a mess! And you are only given the graph of f(x). How do we go about sketching g(x)?

As in any problem in mathematics, let's break it all down to a list of simple steps. We start with a graph of f(x).

1. Let's create a function f1(x) = f(x+1). You might already know that adding a number to the argument of a function is equivalent to 'sliding' its graph by that number to the left (it's quite simple, after all, you make an x=5 give f(5+1)=f(6) - which is to the right of f(5) we call that translation by a vector)

2. f1(x) is not quite like g(x) yet. Let's make f2(x) = f1( |x| ) = f( |x+1| ) . How does that affect the graph? As we know |x| = x, for positive x's and |x| = -x, for negative x's. So to sketch f2(x), we take the part of the graph of f1(x) which is to the right of the y-axis (where x is positive), and on the left half, we need to draw a mirror image of the right half. We're one step closer to g(x).

3. Next we need to include the number -3 in the argument. Let f3(x) = f2(x-3). Which means another shift along the x-axis. This time we move the graph of f2(x) to theright, by 'amount' 3. 

4. Now we are 'outside' f3(x) - we are not going to play with the argument anymore. Which means that now we will change the graph in the y-axis only. The remaining steps include multiplying the function by 4. This means that everywhere on the graph, the function is 4 times farther away from the x-axis (from 0). So sketch f4(x) = 4*f3(x), 4 times taller and steeper.

5. Then there is the 5 left to add. If we add a number to the function, it means that it goes up in the y direction by 5, everywhere. So let's shift the graph upwards. f5(x) = f4(x) + 5

6. We are nearly done now. The last thing to do is the absolute value. Once again, for all positive values, it stays as is, and for all negative values it gets a minus sign. Thus everywhere the function f5(x) has values smaller than 0, we need to 'flip' it upside down (symmetry with respect to the x-axis). f6(x) = |f5(x)|

g(x) = f6(x)

And that is it. You now have an accurate depiction of g(x). It wasn't that complicated after all. You just needed to make incremental changes and step by step move 'from the inside out'.

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