The first problem on the warm-up is really important, and most students will need some explanation about how the graph “bounces off” the axis at the root when the factor is squared, whereas it passes through the axis when the root is not squared. For a more general statement, when a root has an even multiplicity, the graph will bounce of the axis at that point; when a root has an odd multiplicity, the graph crosses through the axis at this point. To use more academic vocabulary, you can introduce the concept that when the multiplicity of the root is 2 (or any other even number) the graph is tangent to the x-axis, whereas when the root has a multiplicity of 1 (or any other odd number) the graph intersects the x-axis.
Another question that students can probably start to answer while looking at these first two graphs is: “If we see that the degree of the polynomial is even, what do we know about the end behavior?” Some students have no idea how to answer this, but it is a good extension question for them to think about as they are creating the graphs.
The second problem relates to yesterday’s lesson. The discussion about 0 x-intercepts is worth having with small groups, though it takes some time. For students who are struggling with the basic idea that the number of x-intercepts cannot be greater than the degree, it is fine to skip this discussion. Otherwise, you can ask students why it would be possible to have 0 x-intercepts with an even degree but it is not possible with an odd degree. This relates to the previous discussion about end behavior. It also previews the idea of a vertical shift, which some students will come up with on their own and other students will need to have explained to them.
The important thing is that students understand that the number of x-intercepts cannot be greater than the degree, and this part of the warm-up gives students another chance to articulate that idea.
The product of several numbers is 0 as long as any one of the numbers is 0, no matter what other number you multiply this product by. Consider the equation . a, b or c have to be 0 in order for this equation to be true. The 4 has no affect on this.
Yes, if the ends of the polynomial go in the same direction (both up or both down), the degree must be even. If the ends of the polynomial go in different directions (one up and one down), then the degree must be odd. This is because either a positive number or a negative number raised to an even power will give a positive output, whereas if you raise a negative number to an odd power, the result is still negative, so the ends will go in different directions.
Yes. If the ends both go up, the coefficient must be positive. If the ends both go down, the coefficient must be negative. If the function increases positively as x increases but decreases to infinity as x approaches negative infinity, the coefficient will also be positive…
No, you need to use a point that isn’t on the x-axis, because the function will have the same x-intercepts no matter what the value of the coefficient is.