SOLUTION: The quadratic function {{{f(x)=ax^2+bx+c}}} has the following characteristics: (i) passes through the point (2,4); (ii) has a maximum value of 6 when x=4; and (iii) has a zero of {

Algebra ->  Functions -> SOLUTION: The quadratic function {{{f(x)=ax^2+bx+c}}} has the following characteristics: (i) passes through the point (2,4); (ii) has a maximum value of 6 when x=4; and (iii) has a zero of {      Log On


   

Question 979388: The quadratic function f%28x%29=ax%5E2%2Bbx%2Bc has the following characteristics: (i) passes through the point (2,4); (ii) has a maximum value of 6 when x=4; and (iii) has a zero of x=4%2B2sqrt%283%29.
Find the values of a,b, and c.
Found 2 solutions by josgarithmetic, Boreal:
Answer by josgarithmetic(39615) About Me  (Show Source):
You can put this solution on YOUR website!
The maximum is a vertex, (4,6) and you can say y=a%28x-4%29%5E2%2B6.

The given zero should be enough for finding value of a. Expect to use or find a%3C0.

y-6=a%28x-4%29%5E2
a=%28y-6%29%2F%28x-4%29%5E2
a=%280-6%29%2F%284%2B2sqrt%283%29-4%29%5E2
a=-6%2F%282sqrt%283%29%29%5E2
a=-6%2F12
a=-1%2F2

Standard Form function is highlight%28f%28x%29=-%281%2F2%29%28x-4%29%5E2%2B6%29.
Simply do the simplification (multiplication...) to put into the general form to identify b and c.
You already know a.

Answer by Boreal(15235) About Me  (Show Source):
You can put this solution on YOUR website!
f(4)=6
If there is a zero at 4+2 sqrt (3), there is a zero at 4 - 2 sqrt (3)
a*4+b*2+c=4
4a+2b+c=4
16a+4b+c=6
Eliminate c
4a+2b+c=4
-16a-4b-c=-6
-12a-2b=-2
12a+2b=2
6a+b=1
But -b/2a=4, so -b=8a
-2a=1
a=(-1/2)
b=4
first equation -2+8+c=4; c=-2
second equation -8+16+c=6; c=-2
(-1/2)x^2+4x-2=f(x)
quadratic formula:
{-4 +/- sqrt (16-4)}/-1
roots are 4+/- sqrt (12); sqrt (12=2 sqrt (3))
a= -1/2
b=4
c= -2
graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C%28-1%2F2%29x%5E2%2B4x-2%29