A teenager who is 5 feet tall throws an object into the air. The quadratic function f(x)=−16x2+64x+5 %C2%A0is where f(x) is the height of the object in feet and x %C2%A0is the time in seconds. The function in vertex form is (1 point) Responses f(x)=−16(x−2)2−59 f of x is equal to negative 16 times open paren x minus 2 close paren squared minus 59 f(x)=−16(x−2)2+5 f of x is equal to negative 16 times open paren x minus 2 close paren squared plus 5 f(x)=−16(x−2)2%C2%A0−11 f(x)=−16(x−2)2%C2%A0−11 f(x)=−16(x−2)2+69
The correct answer is:
f(x)=−16(x−2)2+5
This is the vertex form of the quadratic function, where the vertex is located at (2, 5).
The object hits the ground after about%C2%A0 %C2%A0seconds.
To find the time at which the object hits the ground, we need to set f(x) (the height of the object) equal to 0, since the object hits the ground when the height is 0.
So, we have:
0 = -16x^2 + 64x + 5
To solve this quadratic equation, we can factor or use the quadratic formula.
Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = -16, b = 64, and c = 5. Substituting these values into the quadratic formula, we have:
x = (-64 ± √(64^2 - 4(-16)(5))) / (2(-16))
x = (-64 ± √(4096 + 320)) / -32
x = (-64 ± √(4416)) / -32
x = (-64 ± 66.4) / -32
We can simplify this further:
x ≈ (-64 + 66.4) / -32 ≈ 0.075 seconds
x ≈ (-64 - 66.4) / -32 ≈ 4.075 seconds
Therefore, the object hits the ground after about 0.075 seconds and 4.075 seconds.