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

please help

Answer 1

A.

f is a quadratic function, which means it's graph is a parabola.

Notice that the coefficient of $x^{2}$ is negative, so the parabola opens downwards.

the x-coordinate of a parabola is always determined by the formula: $-\frac{b}{2a}$

where a is coefficient of the $x^{2}$ term, and b is the coefficient of the x term.

Thus, x-coordinate of the vertex of the graph of f is :

$-\frac{b}{2a}=-\frac{16}{2(-1)}=8$

the y-coordinate of the vertex is f(8)=-8*8+16*8-60=4.

The vertex is (8, 4).

This means that the maximum daily profit is when exactly 8 candles are sold.

B.

The x-intercepts are the values of x such that f(x)=0,

so to find these values we solve:

$- x^{2} +16x-60=0$


$x^{2}-16x+60=0$

complete the square:

$x^{2}-2*8x+64-64+60=0$

$(x-8)^{2}-4=0$

$(x-8)^{2}= 2^{2}$

so x-8=2      or x-8=-2

the roots are x=10 and  x=6, are the roots.

This means that when the shop sells exactly 6 or 10 candles, it makes no profit.