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. Determine the vertex, and explain what it means in the context of the problem. (6, 10); The vertex represents the maximum profit. (6, 10); The vertex represents the minimum profit. (8, 4); The vertex represents the minimum profit. (8, 4); The vertex represents the maximum profit.

Question

Mathematics

Kylan Bridges

31 December, 17:44

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. Determine the vertex, and explain what it means in the context of the problem. (6, 10); The vertex represents the maximum profit. (6, 10); The vertex represents the minimum profit. (8, 4); The vertex represents the minimum profit. (8, 4); The vertex represents the maximum profit.

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Answers (1)

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Kianna Casey · Answer 1

A.

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

Notice that the coefficient of is negative, so the parabola opens downwards.

the x-coordinate of a parabola is always determined by the formula:

where a is coefficient of the term, and b is the coefficient of the x term.

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

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:

complete the square:

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.