Question

a ball is launched from a sling shot. its height, h(x), can be represented by a quadratic function in terms of time, x, in seconds. after 1 second, the ball is 110 feet in the air; after 2 seconds, it is 200 feet in the air. complete the height function, h(x), for this situation. h(x) = enter the correct answer.

Explanation:

Step1: Define quadratic form

Let $h(x) = ax^2 + bx + c$. At $x=0$, $h(0)=0$, so $c=0$. Simplify to $h(x)=ax^2+bx$.

Step2: Set up equations from given data

For $x=1, h(1)=110$: $a(1)^2 + b(1) = 110 \implies a + b = 110$
For $x=2, h(2)=200$: $a(2)^2 + b(2) = 200 \implies 4a + 2b = 200$

Step3: Solve the system of equations

First, simplify the second equation: divide by 2: $2a + b = 100$
Subtract the first equation: $(2a + b) - (a + b) = 100 - 110$
$\implies a = -10$
Substitute $a=-10$ into $a + b = 110$: $-10 + b = 110 \implies b = 120$

Step4: Write the final function

Substitute $a=-10$ and $b=120$ into $h(x)=ax^2+bx$.

Answer:

$h(x) = -10x^2 + 120x$