Numerical Values for the last three sections of this page are provided for guidance. Student answers will vary.
Vertex form of a Quadratic Model
Vertex Form: f(x) = a(x − h)2 + k |
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1 We’ve determined that the optimum bat angle is around 30 degrees. What variable in the equation should we replace with 30?
2 What y-coordinate of the vertex (vertical shift) would best match the shape of the curve?
3 Does it make sense for a to be negative or positive for this curve?
Fitting the Model Visually in Desmos
For this section, you’ll need to have Exploring Quadratic Functions(Desmos) open to Slide 3: Fitting a Model: Bat Angle v Hit Distance (Curveballs).
4 Using your thinking about the values of a, h, and k from above, adjust the sliders to fit a quadratic model to the data. Continue adjusting the sliders until you’ve landed on the best model you can. Record your values for a, h, and k below.
a: h: k:
5 Using the values of a, h, and k you decided on in the Desmos file, define your quadratic function below in Pyret notation.
fun curve(x): ( * sqr(x - )) + end
Testing how Good the Model is in Pyret
Return to your copy of the Aaron Judge Starter File, adjust the definition for curve(x) on line 42, and click "Run".
6 Use fit-model to fit curve(x) to the curve-table data. What S-value did you get?
Hint: If you forgot the contract for fit-model, look it up in the Contracts Pages!
7 The S-value for the optimal linear model was about 104 feet.
My quadratic model should do a little better, much better, worse, much worse, an equally good job at predicting hit distances from bat angles.
What does this model actually mean?
After experimenting, I came up with a quadratic model for this dataset showing that x-variable is correlated to y-variable.
The vertex of the parabola drawn by this model is a minima or maxima? at about (x, y), which means that
Before this point, as bat angle increases, hit distance . After this point, as the angle increases hit distance
The error in the model is described by an S-value of about Sunits. I strongly agree, agree, disagree, strongly disagree that this model is a good fit considering that y-variable in this dataset range from lowest y-value to highest y-value.
These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, 1738598, 2031479, and 1501927). 
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