Graphing cubic function transformations
About this video
Starting from f(x) = x^3, the video sketches two transformations. y = -2f(x) flips the graph over the x-axis and stretches it vertically by 2, so the point (1, 1) becomes (1, -2). y = f(x + 4) shifts the whole graph 4 units to the left.
You'll be able to
- describe how y = -2f(x) changes the graph of y = f(x).
- explain why replacing x with x + 4 shifts a graph to the left.
- sketch transformed cubic graphs using key points.
Key words
- reflection
- Flipping a graph over a line, such as the x-axis.
- vertical stretch
- Multiplying all the outputs by a number bigger than 1, making the graph steeper.
- horizontal shift
- Sliding a graph left or right without changing its shape.
- cubic function
- A function with x^3 as its highest power.
Transcript
- [Lecturer] So what we have right over here, this is the graph of y is equal to f of x, where f of x is equal to x to the third. What I wanna do in this video is transform this f of x a little bit and see what our graph should look like based on that. So the first transformation I want to graph, I want to graph y is equal to negative, let's go with negative two times f of x.
What should that look like? Well, there's two things to think about. Just this negative would flip it over. If I just said y is equal to negative f of x, we would, one way to think about it is it would be reflected over, it would be flipped over the x axis. So it would, if we just did negative f of x, it would look something like, I'm trying my best to hand draw that.
It would look something like that. But we don't just have a negative, we also have a two here, negative two. So it's also going to stretch it in the vertical direction. One way to think about it, where your function would've equaled one, now it's going to be equal to negative two, not just negative one.
So, and that actually might be an easy way to think about this, 'cause if you think about cubing things, the values go up very fast. So instead of a positive one here, you're now going to be at a negative two. Similarly in at a negative one, instead of the negative one cube being negative one, you now have negative one cubed times negative two.
So it would put you there. So our new graph is going to look something like this. We're going to be going up much, much, or twice as fast I should say. And in the opposite, I guess you could say, flipped over the x axis. So it's going to look something like this.
So it's gonna get steeper faster. In fact, I should probably, yeah, that's reasonable for now. It's gonna get steeper than faster because of this two and the negative flips it over. So now let's just do an example where we shift our original graph. So let's imagine another graph y is equal to f of x plus four.
What would that look like? Well, one thing to remember is if you replace your x with an x plus four, this is going to shift everything to the left by four. I always used to get confused if I'm saying plus four here, wouldn't it shift everything to the right? But you then have to remember what used to with the value for function, you would get at x equals zero, you're now going to get it, x equals negative four.
Negative four plus four is zero. So everything is going to shift here. So you're gonna have your original graph shifted to the left. So it's going to look like, it's going to look like this. And I'm just picking some values where I know the value. So it's going to look something like this now.
All I did is shift it to the left by four, the original f of x to the left by four and we're done.
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