Polynomial factors and graphs — Harder example
About this video
A graph with x-intercepts at -4, 0, and 3 is matched to a polynomial equation. Each intercept gives a factor: x for zero, x plus four for negative four, and x minus three squared for positive three. The sign of the leading term is then chosen by checking how y behaves as x grows positive and negative, which rules out the wrong choices.
You'll be able to
- connect x-intercepts to factors of an equation.
- use end behavior to decide between candidate sign patterns.
- verify a candidate equation against the shape of a graph.
Key words
- X-intercept
- A point where the graph crosses the x-axis, where y equals zero.
- Factor
- A term such as (x + 4) that makes the expression zero when x equals a particular value.
- End behavior
- How y behaves as x becomes very large positive or very large negative.
- Squared factor
- A factor raised to the second power, which keeps the sign the same on both sides of its root.
- Verification
- Checking a candidate equation against more than one feature of the graph before choosing it.
Transcript
- [Instructor] We're asked which of the following could be the equation of the graph above. So pause this video and see if you can work through this before we do it together. All right, so first of all, let's just think about where the graph intersects the X axis.
We can see it does that at X equals negative four. It does that at X equals zero. And it does that at X equals positive three. So we would expect to see an expression that's equal to zero at these three places. So first of all, X equals zero. You could see if X equals zero.
All of these will become zero, 'cause zero times anything else is zero. So they all meet that one. See, X equals negative four. In order for it to become zero at X equals negative four, we'd want to see an X plus four when we factor things out. We see an X plus four here, we see an X plus four here.
These second two, they have X minus four. So we don't like those. So we can immediately rule those out. And then for X equals three to make the entire expression equal to zero, you'd need to have an X minus three. And we can see we have an X minus three and an X minus three.
So now we have to decide what makes more sense to have a negative X out here or a negative X squared. Well, let's think about what would happen in either case as X increases. As X increases, as X becomes larger and larger positive values, you take the negative of that, and it makes sense in this A choice that the value would get more and more negative.
That's also the case in choice B. As X increases, you square it. You'll get even a more positive value. And then you take the negative of that, it should go down. So both choice A and B is consistent with this behavior of as X becomes more positive, Y is becoming more and more negative.
Now let's look on the other hand. What about when X becomes more and more negative? Well, when X becomes more and more negative in choice A, when you take the negative of a negative you would expect this part to become more and more positive. So you would expect for choice A, that actually the Y values would increase as X becomes more and more negative, which clearly isn't happening.
So we can rule that one out, as well. But let's just verify that B works. So when X becomes more and more negative, you take the square of that. So you're going to get a positive value, but then you take the negative of that. So it's going to become, so Y will keep decreasing and becoming more and more negative, which is exactly what we see in this graph.
So we're liking choice B.
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