Question

Pls help I'll follow and medal

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

For any quadratic functions, ax² + bx + c, we can easily find the x-coordinate of its vertex through the use of the formula below.


`x_(vertex) = (-b)/(2a)`

where a and b are the coefficients of the function. For f(x), we have a = -1 and b = -16. Thus, we have


`x_(vertex) = (-16)/(2(-1))`

`x_(vertex) = 8`

Since we now have the x-coordinate of the vertex, we can just easily substitute 8 into f(x) as shown below.


`y_(vertex) = f(8) = -(8)+16(8)-60`

`y_(vertex) = 60`

Thus, the vertex of f(x) is (8, 60). The vertex of the function shows the maximum value that the function can reach. For this particular case, the vertex represents the maximum profit that the shop owner could earn.

The x-intercepts of the function are the values of x when f(x) is zero. By equation f(x) with zero, we can solve the quadratic equation to find the x-intercepts as shown below.


`-x^(2)+16x-60 = 0`

`x^(2)-16x+60=`

`(x-6)(x-10) = 0`

`x = 6`

`x=10`

From this, we can see that the x-intercepts are (6, 0) and (10, 0). We just discussed that the x-intercepts are the values of x when y is zero. So, for this case it means that the shop does not earn a profit if they sell 6 or 10 candles. Basically, the x-intercepts represent the number candles sold if the shop wants to break even.