Question

two balls are tossed up into the air. the function $f(x) = -4.9x^{2} + 14.7x + 0.975$ models the height of ball a. the height of ball b over time is shown in the table to the right. which ball reaches a greater height? how much greater? explain how you can answer without graphing either function.

time (s) $x$

height (m) $g(x)$

----

----

0

0.725

1

10.525

1.5

11.75

2

10.525

the vertex of ball a can be calculated as the point $square$. using $\boldsymbol{

abla}$ the vertex of ball b is the point $square$. since the y-coordinate of the vertex of ball $\boldsymbol{
abla}$ is greater than that of ball $\boldsymbol{
abla}$ ball $\boldsymbol{
abla}$ has a greater max height of $square$ meters.(simplify your answers. type integers or decimals.)

Explanation:

Step1: Find vertex x of Ball A

For quadratic $f(x)=ax^2+bx+c$, $x=-\frac{b}{2a}$
$x=-\frac{14.7}{2(-4.9)} = 1.5$

Step2: Find vertex y of Ball A

Substitute $x=1.5$ into $f(x)$
$f(1.5)=-4.9(1.5)^2+14.7(1.5)+0.975 = 12.0$

Step3: Identify vertex of Ball B

Symmetry of table values: vertex at $x=1.5$
From table, $g(1.5)=11.76$

Step4: Compare max heights

Subtract heights: $12.0 - 11.76 = 0.24$

Answer:

The vertex of Ball A can be calculated as the point $(1.5, 12.0)$. Using symmetry the vertex of Ball B is the point $(1.5, 11.76)$. Since the y-coordinate of the vertex of Ball A is greater than that of Ball B, Ball A has a greater max height of 12.0 meters, which is 0.24 meters greater than Ball B's max height.