Functions and Their Mothers

Betcha didn’t know that functions have mothers.  Well.  They do.

I know what you’re thinking…”who cares”…?  Right?

Well I’ll tell ya why you should care, one day you’re going to be stranded on a desert island without your graphing calculator or the internet and the local pirates are going to give you a choice… walk the plank or graph f(x)= -2(x+1)^3+5.  Yup.  It could totally happen.

Alright maybe that wouldn’t happen, but probably sometime you’ll have to graph f(x)= -2(x+1)^3+5 (or some other function like it) in your math class.  Here’s how:

First, you have to figure out what your mother function is.  You’ll probably have a few examples of mother functions in your math textbook… x squared… square root of x… some number to the power of x… that sort of thing.  The mother function in our case is x cubed… the graph of x cubed looks like this:
transformations 1

Ok cool, but x cubed is not the graph we’re looking for.  When we stick and x into f(x), the first thing we do is add 1.  So let’s graph the function f(x)= (x+1)^3 using the graph of x cubed.  This shifts the previous graph one to the left, like this: transformations 2

That’s nice, but that’s still not the graph we’re looking for.  After we add 1 and cube x, we multiply by 2, so let’s graph the function f(x)= 2(x+1)^3 using the previous graph.  Multiplying by 2 will stretch the graph vertically by a factor of 2, like this:
transformations 3

That’s all fine and good, but we ain’t done yet.  After we add 1 and cube x and multiply by 2, we have to multiply by -1.  This’ll flip the previous graph across a vertical line, so the graph of f(x)= -2(x+1)^3 will look like this:
transformations 4
The last thing we have to do is add 5… this shifts the previous graph up five units…transformations 5
And finally!  That’s the graph of f(x)= -2(x+1)^3+5!  Woo-hoo!  Just remember to start with the graph of the mother function and then modify the graph in the order of modifications to x and you won’t have to walk the plank!

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