A basketball is thrown from the roof of a 50 foot building. It reaches a maximum height of 86 feet 1.5 seconds after the ball is thrown. Write a function representing the problem situation in vertex form, f(x)=a(x-h)^2+k, and standard form, f(x)=a^2+bx+c.

Step-by-step explanation:

The basketball reaches a maximum height of 86 feet after 1.5 seconds, so the vertex is (1.5, 86).

f(x) = a(x − 1.5)² + 86

We know the initial height is 50 feet, so plug in the point (0, 50) to find the value of a.

50 = a(0 − 1.5)² + 86

-36 = 2.25a

a = -16

f(x) = -16(x − 1.5)² + 86

Simplify to get the standard form.

f(x) = -16(x − 1.5)² + 86

f(x) = -16(x² − 3x + 2.25) + 86

f(x) = -16x² + 48x − 36 + 86

f(x) = -16x² + 48x + 50

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!

step-by-step explanation:

a child is 12 years old, and his father is 32 years older. in how many years will the age of the father be 2 times the age of the child?

******************

this needs to be the answer format, or else your answer is incomplete.

if the number of years is x, then the equation is

in __ years