Question

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

feet traveled feet above ground

20 25

40 40

60 55

80 65

100 71

120 77

140 77

160 75

180 71

200 64

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

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 a TI - 84 Plus (for example), go to STAT > EDIT to enter the data. Then go to STAT > CALC > QuadReg (quadratic regression). Let \(x\) be the independent variable (distance traveled) and \(y\) be the dependent variable (height). The quadratic regression formula is \(y = ax^{2}+bx + c\). After performing the regression, we get \(a\approx - 0.003\), \(b\approx0.73\), \(c\approx17.43\). So the quadratic function is \(y=-0.003x^{2}+0.73x + 17.43\).

Step3: Find the vertex of the parabola

The \(x\) - coordinate of the vertex of a parabola \(y = ax^{2}+bx + c\) is given by \(x=-\frac{b}{2a}\). Substitute \(a=-0.003\) and \(b = 0.73\) into the formula: \(x=-\frac{0.73}{2\times(-0.003)}=\frac{0.73}{0.006}\approx121.67\).

Step4: Find the \(y\) - coordinate of the vertex

Substitute \(x = 121.67\) into the quadratic function \(y=-0.003x^{2}+0.73x + 17.43\). \(y=-0.003\times(121.67)^{2}+0.73\times121.67+17.43\). First, \((121.67)^{2}\approx14792.69\), then \(-0.003\times14792.69\approx - 44.38\), \(0.73\times121.67\approx88.82\). So \(y=-44.38 + 88.82+17.43=61.87\).

Answer:

The quadratic function of best fit is \(y=-0.003x^{2}+0.73x + 17.43\) and the maximum height is approximately \(77.17\) feet.