This illustrative mathematics activity from OER Commons would be a good opening activity to exploring characteristics of quadratic functions.  This activity provides an opportunity to compare solutions, factors, zeros, equations, and graphs. https://tasks.illustrativemathematics.org/content-standards/tasks/388

As students learn about graphs of quadratic functions using the Polygraph on quadratic functions would provide a great opportunity for students to practice vocabulary.  This guess who style game requires students to identify similarities and differences between parabolas.  https://teacher.desmos.com/polygraph-parabolas

Students could then complete this google draw to compare graphs, equations, factors, and solutions.  https://docs.google.com/drawings/d/1FQztfTHtVct_dNIUYWUQ3fAHx4HQ1BRgf3f0x6b64zE/copy

You could then have students work through this desmos activity that relates the graphs and solutions to the discriminant.  The activity was created by Mrs. Belman https://teacher.desmos.com/activitybuilder/custom/5c73b6ed8917474b9a242eb5