To determine the vertex of the given quadratic function [tex]\( f(x) = -x^2 + 16x - 60 \)[/tex], we need to follow the methods typically used for finding the vertex of a parabolic equation of the form [tex]\( ax^2 + bx + c \)[/tex].
Step-by-Step Solution:
1. Identify the coefficients:
The quadratic function is given in the standard form [tex]\( ax^2 + bx + c \)[/tex]. From [tex]\( f(x) = -x^2 + 16x - 60 \)[/tex], we have:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = 16 \)[/tex]
- [tex]\( c = -60 \)[/tex]
2. Calculate the x-coordinate of the vertex:
The formula for finding the x-coordinate of the vertex of a parabola [tex]\( ax^2 + bx + c \)[/tex] is given by:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
x = -\frac{16}{2 \cdot (-1)} = -\frac{16}{-2} = 8
\][/tex]
3. Calculate the y-coordinate of the vertex:
Substitute the x-coordinate back into the function to find the y-coordinate. That is:
[tex]\[
f(8) = -(8)^2 + 16 \cdot 8 - 60
\][/tex]
Simplifying:
[tex]\[
f(8) = -64 + 128 - 60 = 4
\][/tex]
4. Determine the vertex:
Hence, the vertex of the given quadratic function is [tex]\( (8, 4) \)[/tex].
5. Interpretation of the vertex:
In the context of the problem, the function [tex]\( f(x) \)[/tex] represents the daily profit in dollars for selling [tex]\( x \)[/tex] candles.
- The x-coordinate of the vertex (8) represents the number of candles sold that maximizes the profit.
- The y-coordinate of the vertex (4) represents the maximum profit in dollars.
Since the coefficient of the [tex]\( x^2 \)[/tex] term ([tex]\(-1\)[/tex]) is negative, the parabola opens downwards, indicating that the vertex represents a maximum point.
Therefore, the correct interpretation and vertex are:
- Vertex: [tex]\((8,4)\)[/tex]
- The vertex represents the maximum profit.
The correct answer is:
- [tex]$(8,4)$[/tex] : The vertex represents the maximum profit.
