Teach 180: Connecting Representations (Day 77)

One of these polynomials only has real zeros and the other one only has complex zeros.  Which is which and how do you know?

F(x) = x² + 4                                          G(x) = x² - 6x + 8

I asked this question of my PreCalculus students today.  I was curious if they would focus on an algebraic approach or a graphical approach.  My bet was on an algebraic approach.  They took a minute or two to discuss this at their tables and all groups reached the same conclusion based on, no shock, an algebraic approach.  They set each function equal to zero and solved the resulting equation.

Because I think it is important for students to connect various representations, we graphed the two functions in Desmos.  F(x) is the red parabola and G(x) is the blue parabola.  

I asked, "Why does the red parabola have complex zeros?  How could you tell that from the graph?" One student answered, "It has complex zeros, because the vertex is on the y-axis."  This was not what I was expecting.  But rather than throwing that back at the class to see what they would do with it, either confirm or refute it, I entered y = x² - 1 into Desmos to show a parabola with a vertex on the y-axis and two real zeros.  What I should have done was had the students use Desmos themselves to either prove or disprove the student's statement.  Clearly, I understood the connection between the algebraic and graphical representations, but did my students?  I have some thoughts about how I will assess this when I see my students on Friday.  That will be for another blog entry.