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Video: Sketching Curves from their Roots

Bethani Gasparine

Which of the following is the graph of 𝑓(𝑥) = −(𝑥 − 1)²(𝑥 + 1)²?


Video Transcript

Sketch the graph of 𝑓 of 𝑥 equals negative 𝑥 minus one squared 𝑥 plus one squared.

So before we sketch the graph, let’s look at some features. If we would multiply all of this out, the leading coefficient would be a negative 𝑥 to the fourth. So it would have a degree of four, so it will be an even function with a negative leading coefficient.

So if it’s even with a negative leading coefficient, the arrows will go downwards. Looking at the factors, if we would set those equal to zero, we will know where we cross the 𝑥-axis. So we will cross out one and negative one; however, we also need to look at the multiplicity of these factors.

Since each of them are squared, they have an even multiplicity. So instead of crossing through at that point, it will actually bounce off of that point. So at one and negative one, we won’t go through them; we will bounce off of them.

And we know our arrow should be facing downwards. So as I begin to draw this, instead of going through, we will bounce off, and same with the other one, and we go back down. So this would be the sketch of that graph.