The table below shows the height (in feet) of a ball at several times after it was launched. Use a graphing calculator to perform quadratic regression on the data. Write the quadratic equation of best fit in the form $y=a x^2+ b x+c$. Round each coefficient to the nearest tenth.

Seconds Height (feet)

1 20

2 30

3 44

4 62

5 84

6 110

$y = \_\_ x^2 + \_\_ x + \_\_$

To determine the quadratic equation of best fit $y = ax^2 + bx + c$, we start by identifying the given data points: $(1, 20), (2, 30), (3, 44), (4, 62), (5, 84),$ and $(6, 110)$. We use the method of least squares regression to find the coefficients $a, b,$ and $c$ that minimize the sum of the squares of the vertical deviations of the data points from the parabola. This provides the best fit model for the given set of observations.

Performing Quadratic Regression

We perform a quadratic regression on the given data points. By calculating the sums required for the normal equations—specifically $\sum x_i, \sum x_i^2, \sum x_i^3, \sum x_i^4, \sum y_i, \sum x_i y_i,$ and $\sum x_i^2 y_i$—we can solve for the parameters $a, b,$ and $c$. The regression process determines these parameters as follows:
$a \approx 2.0$
$b \approx 4.0$
$c \approx 14.0$
These values provide the best fit curve representing the relationship between the time and the height of the ball.

Formulating the Final Equation and Conclusion

Substituting the calculated coefficients back into the standard quadratic form $y = ax^2 + bx + c$, we obtain the equation of best fit. Since the coefficients are already consistent with integers, rounding them to the nearest tenth as requested results in:
$a = 2.0$
$b = 4.0$
$c = 14.0$
Thus, the final quadratic equation is $y = 2.0x^2 + 4.0x + 14.0$.

Examples

Quadratic regression is widely used in physics and engineering to analyze the motion of objects under constant acceleration, such as gravity. For example, when an object is thrown into the air, its height over time follows a parabolic path. By measuring the height at various time intervals, scientists and engineers can use the quadratic model $h(t) = at^2 + bt + c$ to predict the object's peak height, the time at which it hits the ground, and its velocity. This tool is fundamental in fields ranging from sports science, where researchers analyze the trajectory of a ball, to ballistics and space exploration.