Graph a polynomial with transformations
About this video
The video graphs g(x) = (1/2)(x - 3)^3 + 5 by treating it as a transformation of the parent function f(x) = x^3. It first scales the cubic by 1/2, then shifts it right 3 and up 5, tracking key points such as (0, 0), (1, 1) and (2, 8) as they move.
You'll be able to
- identify the parent function and the scale factor and shifts in g(x) = a(x - h)^3 + k.
- explain how multiplying x^3 by 1/2 changes the graph.
- move key points of a cubic to sketch a transformed graph.
Key words
- parent function
- The simplest function of a family, like f(x) = x^3, that other graphs are made from by transforming it.
- scaling
- Multiplying a function by a constant, which stretches or squashes the graph vertically.
- shift
- Sliding a graph left, right, up or down without changing its shape.
- transformation
- A change to a graph such as a shift or a scaling.
Transcript
- [Instructor] What we wanna do in this video is try to graph the function g of x is equal to one half times x minus three to the third power plus five. So if you have some graph paper in front of you, have a go at it. Or even if you don't, just think about how you would try to graph this.
Alright, now let's work on this together. So this is really a transformed version of a function. Let's call it f of x is equal to x to the third. And the reason why I know it's a transformed function is if I were to put a constant in front of x to the third. So if I were to say this is a x to the third, that's just scaling it and that's analogous to the what this one half is.
And then if I were to shift it to the right by three, then I would replace the x with x minus three to the third, and then if I were to shift it up, that's what the plus five is. So g of x is just f of x where I'm shifting it to the right by three and up by five and I'm scaling it by one half.
So let's just work on that a little bit. So what does f of x equal x to the third look like? Well, f of zero is zero. F of one is one. F of negative one is negative one, 'cause negative one to the third is negative one. And then f of two is, let's see, 1, 2, 3, 4, 5, 6, 7, 8, be right over there.
On this side it kind of falls off. Let's see, negative two would be negative eight. So negative two, negative four, negative six. You can't even see where I am writing now, I'm off the screen, but the graph is going to look something like this. This is my best hand-drawn version of what it would look like.
So you could view this as almost the parent function and now we just need to transform it. And we can do this really... Well, we could do the shifting in either order, but let's scale it first. So if we were to do one half times this, so if we were to say one half, so lemme write another function here, h of x is equal to one half times x to a third, well, h of zero is still going to be zero, but h of one, instead of it being one, is not going to be one half.
You have to multiply by this one half here. Similarly, h of negative one, instead of being negative one, is gonna be negative one f. Half h of two, instead of it being eight is not going to be half of that. It's now going to be four. H of negative two, instead of being negative eight is now going to be negative four.
So h of x would look something like this. H of x is going to look like this. One way to think about it is x increases, it grows slower, and as x decreases, it decreases slower. That's what that one half does. And now we want to shift it by to the right by three.
So if we take, what was this point zero-zero, we shift it to the right by three, so we go there, so that's what that does, and then we shift it up by five, that's what that plus five does for us, so 1, 2, 3, 4, 5, what was at zero, zero will now go there and what was at one, one half is now going to, will go to the one to the right and then half up and then one to the left and half down.
That's analogous to this point right here. And then what was two to the right and four up, well, once again, from this point we go two to the right and 1, 2, 3, 4 up and two to the left and 1, 2, 3, 4 down, and the graph is going to look something like this. And you bear in mind I'm just hand-drawing it so it's not going to be a perfect drawing.
But there you have it. That is my best hand-drawn version of g of x, right over there.
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