Experience First

In this lesson students build on their knowledge of quadratics by connecting the different forms and noticing how each form highlights a distinct feature of the parabola. Students will work through a Desmos activity which can be found here as they follow along on the activity page. Use the pacing feature on Desmos to restrict students to only slides 1-7. Prepare for a debrief of question 5 by snapshotting student responses on slide 7. Be sure you showcase at least one student answer that used factored form, and if time permits, a response that used vertex form. We recommend using the “anonymize” feature so that students can discuss ideas without focusing on who produced that idea. The driving question for the first part of the activity should be how one can know what the vertical stretch or shrink is (the “a” value). Many students will have used guess-and-check and it’s important to highlight this as a valid strategy but one that might be less efficient without the capacities of Desmos readily available. The margin notes in question 5 formalize the thoughts that students should have started to notice in the activity.

Once the idea of how to find the a-value has set-in for students, open up the rest of the activity. Tell students their new challenge is to write 2 equations for each parabola. Encourage group discussion as they try out various equations. Continue to ask “why did you choose to write the equation in that form?” and “what features of the graph are you able to see in this equation?”.

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Formalize Later

The idea of the y-intercept being related to the zeros of a function might be new for students. Two different approaches can be used to demonstrate this. You can have students consider what it would look like to expand a(x-p)(x-q) and notice how only the final product (p*q) would contribute to the constant term. This term then gets multiplied by “a” to create standard form and students should know that the constant term of standard form is the y-intercept (since all the other terms “go away” when x=0). The other approach is to simply plug in x=0 and know that this should equal the y-intercept, and then solve for a by dividing.

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In the Important Ideas, emphasize to students that given one form of a quadratic it is always possible to convert to another form. Converting from standard form to vertex form will be covered in tomorrow’s lesson.