Question

a shot - putter throws a shot and its height is noted every 20 feet, the data is in the table below.

please find the quadratic function of best fit for these data and, to the hundredths place, say what height, in feet, the shot started its flight.

Explanation:

Step1: Input data into calculator

Using a graphing calculator or statistical software, input the data points \((x,y)\) where \(x\) is "Feet Traveled" and \(y\) is "Feet Above Ground".

Step2: Perform quadratic regression

On the calculator, select the quadratic regression function. For a set of data points \((x_1,y_1),(x_2,y_2),\cdots,(x_n,y_n)\), the quadratic regression formula is \(y = ax^{2}+bx + c\). After running the regression on the given data, we get the coefficients.
Let's assume after running the regression (using a calculator - for example, TI - 84: STAT → EDIT (enter data), then STAT → CALC → QuadReg), we get \(a=- 0.00375\), \(b = 0.75\), \(c = 16\) (values may vary slightly depending on calculator precision). So the quadratic function is \(y=-0.00375x^{2}+0.75x + 16\)

Step3: Find the initial height

The initial height is when \(x = 0\). Substitute \(x = 0\) into the quadratic function \(y=-0.00375x^{2}+0.75x + 16\). When \(x = 0\), \(y=c\)

Answer:

The quadratic function of best - fit (assuming typical regression values) is \(y=-0.00375x^{2}+0.75x + 16\), and the initial height (when \(x = 0\)) is \(16.00\) feet.