Ask the students to describe what are the two functions that create this curve (Parabola and a line).
Using prior knowledge have students graph the maximum point on the parabola, and use the dot where the yellow bird took off at a tangent (B), to create the equation of the parabola. Have students graph the parabola in geogabra, overtop the picture, to ensure all calculations have been done correctly.
Next, ask students how to determine the equation of line.
We will need either a) two points or b) a slope and a point. Both of which is impossible, without the use of the tangent button in Geogabra. I explained, we can calculate this without that button!
Have students pick another “point on” the parabola (c), and to calculate the slope between B and C. Ask how do we make this slope more accurate to the slope of the tangent…
Move C closer to B..
Have students move C closer to B, while still staying on the parabola and calculate the new slope. Eventually move C as close as possible to B. The slope should be -0.5 Below is a picture of one of the students’ work
Next using the point B and the slope you can create the equation of line.
From here your choice for extensions: I had students graph the piecewise functions on their calculator and got the following image
Lastly, here is a video on how to determine the slope at a point: