From the student’s work, what can we infer that the student knows?  What is the student thinking? How would you help?

  • It’s interesting that the student understand the coefficients in question determine both the dilation and whether it is reflected in the x-axis, but doesn’t seem to think that one coefficient could handle both of those things. Like, we need one 2 to make sure it points upward, and another 2 to make sure it gets dilated. Also, the fact that there is a 2 attached to the x inside the parentheses makes me think that the student is trying to cope with this as a horizontal dilation rather than as a vertical contraction since it (a) is attached to the x, and (b) has absolute value >1. In fact, maybe that’s why the student is throwing in two coefficients: the first 2 ensures that the parabola points upward (a y-value concern), and the second 2 doubles the x-coordinate (a horizontal stretching concern). The second 2 makes more sense to me. The first one? I would ask why a positive 1 wouldn’t suffice there to make sure it points up, because the kid doesn’t think it’s dilated in both directions…at least it doesn’t seem that way.

  • The student understands that the vertex location indicates what h and k should be and uses the k value correctly. However, the h value is not used correctly. The student understands that the parabola must have a coefficient for the x that is not 1, and that a 2 is somehow involved, however doesn’t use the correct value. The student understands that the leading coefficient will be positive, however for some reason uses 2.

    I would first have the student look at the structure of the graph and see that the roots are easy to see. Therefore, the easiest approach would be to use the y = a(x-1)(x+3) format for approaching the equation. The “a” will need to be solved for by plugging in another known point that is not a root. There are several to choose from. The final answer would be y = .5(x-1)(x+3), which is a valid form to leave it in. However, if the student wanted to find the h,k form, then completing the square with all of the requisite algebra would be needed.