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:
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:
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:
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:
The last thing we have to do is add 5… this shifts the previous graph up five units…
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!