Question

The function f(x) = –x2 + 16x – 60 models the daily profit, in dollars, a shop makes for selling candles, where x is the number of candles sold, and f(x) is the amount of profit. Part A: Determine the vertex. What does this calculation mean in the context of the problem? (5 points) Part B: Determine the x-intercepts. What do these values mean in the context of the problem? (5 points)

Answer 1

The vertex is either the maximum or minimum value
since the leading coefinet is negative (the number in front of the x² term), the parabola opens down and is a maximum


so
A.
a hack version is to use the -b/(2a) form
if  you have f(x)=ax²+bx+c, then the x value of the vertex is -b/(2a)
so
given
f(x)=-1x²+16x-60
the x value of the vertex is -16/(2*-1)=-16/-2=8
the y value is f(8)=-1(8)²+16(8)-60=
-1(64)+128-60=
4
the vertex is (8,4)
so you selll 8 candels to make the max profit which is $4


B.
x intercepts are where the line crosses the x axis or where f(x)=0
solve
0=-x²+16x-60
0=-1(x²-16x+60)
factor
what 2 numbers multiply to get 60 and add to get -16
-6 and -10
0=-1(x-6)(x-10)
set each factor to 0
0=x-6
x=6

0=x-10
10=x
x intercepts are at x=6 and 10
that is where you make 0 profit