You throw a ball straight up from a rooftop. The ball misses the rooftop on its way down and eventually strikes the ground. A mathematical model can be used to describe the relationship for the ball's height above the ground, y, after x seconds. Consider the following data:

Find the quadratic function y = ax2 +bx + c whose graph passes through the given points.

a. y = -12x2 + 80x + 46
b. y = -10x2 + 60x + 64
c. y = -16x2 + 80x + 50
d. y = -16x2 + 100x + 30

To find the quadratic function whose graph passes through the given points, we need to evaluate the "x" values in order to find the "y" values which satisfy the given values (points) in the table.

So, substituting we have:

- A.

[tex]y=-12x^{2} +80x+46\\\\f(x)=-12x^{2} +80x+46[/tex]

Substituting with "x" equal to 1, we have:

[tex]f(1)=-12(1)^{2} +80(1)+46=-12+80+46=114[/tex]

Substituting with "x" equal to 2, we have:

[tex]f(2)=-12(2)^{2} +80(2)+46=-48+160+46=158[/tex]

Now, since evaluating "x" equal to 2, the function does not match with the given table(points), the function A don't pass through the given points.

- B

[tex]y=-10x^{2} +60x+64\\\\f(x)=-10x^{2} +60x+64[/tex]

Substituting with "x" equal to 1, we have:

[tex]f(1)=-10(1)^{2} +60(1)+64=-10+60+64=114[/tex]

Substituting with "x" equal to 2, we have:

[tex]f(1)=-10(2)^{2} +60(2)+64=-40+120+64=144[/tex]

Now, since evaluating "x" equal to 2, the function does not match with the given table (points), the function A don't pass through the given points.

- C

[tex]y=-16x^{2} +80x+50\\\\f(x)=-16x^{2} +80x+50[/tex]

Substituting with "x" equal to 1, we have:

[tex]f(1)=-16(1)^{2} +80(1)+50=-16+80+50[/tex]

Substituting with "x" equal to 2, we have:

[tex]f(2)=-16(2)^{2} +80(2)+50=-64+160+50=146[/tex]

Substituting with "x" equal to 2, we have:

[tex]f(4)=-16(4)^{2} +80(4)+50=-256+320+50=114[/tex]

Hence, since the given values "x" and the function "y" match with the given table, the function whose graph passes throung the given points is:

C- [tex]y=-16x^{2} +80x+50[/tex]

Have a nice day!