In my first post of my quadratic series, __Navigating the Shift - Quadratic Explorations__, I touched on how you can use the fx-991CW and it’s QR functionality and access to ClassPad.net to really help students explore the shifts to a quadratic caused by changing the coefficient of the parent function and add in that constant. The idea of course is for students to develop their own understandings and ‘rules’ instead of relying on disconnected formulas and processes.

When you change the coefficient of the parent quadratic, students can easily see how the graph of the function is impacted. A negative coefficient (e.g. -3x^2) causes the graph to open down, positive to open up (e.g. 2x^2). As the coefficient’s absolute value gets larger, the graph gets “skinnier”, smaller, the graph gets wider (e.g. -4x^2 vs. -.5x^2). Adding a constant (x^2 + c) shifts the graph up (+) or down (-). Being able to visually see and quickly adjust allows students to conjecture, test, and make their own ‘rules’ thus developing their own understandings about the graphs. The hope being they have developed an intuitive understanding of quadratic functions, and if you were to give them the equation -3x^2 + 2, they can, without graphing, tell you the quadratic will point downward, and have a vertex of (0, 2). And even, perhaps, that the graph crosses the x-axis in two points (roots).

I want to continue to explore quadratic shifts by adding in horizontal shifts to provide a more complete picture to quadratics and developing that intuitive understanding of Ax^2 + Bx + C, where without even graphing, there is an idea of what the graph looks like and where key points might be. This provides the conceptual understanding of a quadratic function that hopefully leads to a deeper connection to why and how factoring and the quadratic formula ‘work’.

As always, I find it much easier to show and use technology than to try and explain in words, so the video below demonstrates how to use the fx-991CW scientific calculator to enter the parent functions, then use the QR code to access ClassPad.net via mobile device or, as I am demonstrating, on a computer, and then use the dynamic features of ClassPad.net to manipulate, explore, and make conjectures.

I have again included a PDF from Casio’s Essentials Activities that focuses on the horizontal shift of quadratic functions. You will see a summary, practice problems, and calculator steps, much like those demonstrated in the video.

My last post in this quadratic series will be connecting this conceptual understanding about quadratic functions, shifts, and key points to some real contexts in order to apply mathematics and make it relevant to students.

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