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The graph of function g is a parabola with the vertex located at (5, 9). The parabola also passes through the points (7, 1) and (3, 1). Select an equation in vertex form for the function.
A. g(x) = 2(x – 5)² + 9
B. g(x) = 2(x – 5)² – 9
C. g(x) = –2(x – 5)² + 9
D. g(x) = –2(x + 5)² + 9
The correct answer is C. g(x) = –2(x – 5)² + 9.
Here’s why:
1. The vertex of the parabola is at (5, 9), which fits the vertex form of a parabola: ( g(x) = a(x – h)^2 + k ), where (h, k) is the vertex. Here, h = 5 and k = 9.
2. We need to determine the value of ( a ). The parabola opens downwards because it passes through points (7, 1) and (3, 1) with a maximum point at the vertex. Therefore, ( a ) must be negative.
3. We can use one of the points (7, 1) to calculate ( a ). Plugging (7, 1) into the equation:
[
1 = a(7 – 5)^2 + 9
]
[
1 = a(2^2) + 9
]
[
1 = 4a + 9
]
Subtracting 9 from both sides:
[
-8 = 4a
]
Dividing by 4 gives:
[
a = -2
]
4. Thus, the equation of the parabola in vertex form is:
[