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Answer:
Vertex: (6, 18)
Step-by-step explanation:
Given the quadratic function, m(x) = -2(x - 3)(x - 9):
Perform the FOIL method on the two binomials, (x - 3)(x - 9) without distributing -2:
m(x) = -2[(x - 3)(x - 9)]
Combine like terms:
m(x) = -2(x² - 9x - 3x + 27)
m(x) = -2(x² - 12x + 27)
where: a = 1, b = -12, and c = 27
Since the axis of symmetry occurs at x = h, then we can use the following formula to solve for the x-coordinate (h ) of the vertex, (h, k):
[tex]x = \frac{-b}{2a}[/tex]
Substitute a = 1 and b = -12 into the formula:
[tex]x = \frac{-b}{2a}[/tex]
[tex]x = \frac{-(-12)}{2(1)} = \frac{12}{2} = 6[/tex]
Therefore, the x-coordinate (h) of the vertex is 6.
Next, substitute the value of h into x² - 12x + 27 to find the y-coordinate (k ) of the vertex:
k = x² - 12x + 27
k = (6)² - 12(6) + 27
k = 36 - 72 + 27
k = 18
Therefore, the vertex of the quadratic function occurs at point (6, 18), in which it is the maximum point on the graph.