## Sunday, December 1, 2013

### Axis of symmetry for a quadratic function

My two-year Algebra course is just starting our study of quadratics. So far, we've looked at graphs and tables for quadratic functions and compared them to linear functions and talked about the symmetry of the shape and finding the axis of symmetry from the x-intercepts and finding the reflections of points across the axis of symmetry. I would like students to derive the formula for the axis of symmetry: $x=\frac{-b}{2a}$for the quadratic function $y=ax^{2}+bx+c$. The Exeter Math 1 curriculum does this by having students graph functions in the form $y=ax^{2}+bx$, then generalizing the idea that one of the x-intercepts is zero and the other one is $-\frac{b}{a}$. I created a worksheet that extends on this idea using desmos to graph and having students also derive the formula for the axis of symmetry by averaging the two x-intercepts to get $x=\frac{-b}{2a}$. Then, just like in the Exeter packet, I will have students compare graphs of $y=ax^{2}+bx$ to graphs of $y=ax^{2}+bx+c$ to realize that the axis of symmetry is not changed by adding a c value to the equation. I'm hoping that the logical progression of the graphs will make sense and that the questions are open enough for this to be genuinely based on student thinking and discovery, but with enough structure that students who are not necessarily used to deriving formulas will have something concrete to think about and answer. Any and all feedback welcome!!