Question

The table below shows the height of a ball x seconds after being kicked. A 2-column table with 5 rows. The first column is labeled time (seconds) with entries 0, 0. 5, 1, 2001. 5, 2, 2002. 5, 3. The second column is labeled height (feet) with entries 0, 35, 65, 85, 95, 100, 95. What values, rounded to the nearest whole number, complete the quadratic regression equation that models the data? f(x) = x2 x 0 Based on the regression equation and rounded to the nearest whole number, what is the estimated height after 0. 25 seconds? feet.

Answer 1

The equation of the quadratic model is $f(x) = -16x^2 + 99x+ 6$, and the estimated height, after 0.25 seconds in 19 feet

A quadratic regression equation is represented as:

$f(x) =ax^2 + bx + c$

Where:

$a = \frac{ [ \sum x^2 y * \sum xx ] - [\sum xy * \sum xx^2 ] }{ [ \sum xx * \sum x^2x^2] - [\sum xx^2 ]^2 }$

$b = \frac{ [ \sum xy * \sum x^2x^2 ] - [\sum x^2y * \sum xx^2 ] }{ [ \sum xx * \sum x^2x^2] - [\sum xx^2 ]^2 }$

$c = [ \frac{\sum y }{ n} ] - { b \times [ \frac{\sum x }{ n} ] } - { a * [ \frac{\sum x^2}{ n} ] }$

Using a graphing calculator, we have:

$a = -16.429$

$b= 81.071$

$c = 0.357$

Approximate to the nearest integers

$a = -16$

$b = 81$

$c = 0$

Substitute these values in $f(x) =ax^2 + bx + c$

$f(x) = -16x^2 + 81x$

Substitute 0.25 for x to calculate the estimated height, after 0.25 seconds

$f(0.25) = -16(0.25)^2 + 81(0.25)$

$f(0.25) = 19.25$

Approximate

$f(0.25) = 19$

Hence, the equation of the quadratic model is $f(x) = -16x^2 + 99x+ 6$, and the estimated height, after 0.25 seconds in 19 feet

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