Jawaban:
a = –4, b = 8, and c = 1.
Pembahasan
Fungsi Kuadrat (Quadratic Functions)
Given that the vertex of y = ax² + bx + c is (1, 5), we know that the axis symmetry for this graph is x = 1, which equals to –b/(2a).
Therefore:
–b/2a = 1
⇔ b = –2a ....(i)
From that vertex, we know that y = 5 when x = 1. The value of y is the minimum or maximum value of ax² + bx + c.
Let’s substitute –2a for b, based on eq. (i).
y = ax² – 2ax + c .....(ii)
And then, substitute 5 for y and 1 for x.
5 = a(1²) – 2a(1) + c
⇔ 5 = a – 2a + c
⇔ 5 = –a + c
⇔ c = a + 5 .....(iii)
Substitute a + 5 for c in eq. (ii).
y = ax² – 2ax + a + 5 .....(iv)
The y-intercept point is at (0, 1).
Let’s plug x = 0 and y = 1 into eq. (iv).
1 = a(0²) – 2a(0) + a + 5
⇔ 1 = 0 – 0 + a + 5
⇔ 1 = a + 5
⇔ a = 1 – 5
⇔ a = –4
Plug a = –4 into eq. (i), we get:
b = –2(–4) = 8
And then, plug a = –4 into eq. (iii), we get:
c = –4 + 5 = 1
CONCLUSION
a = –4, b = 8, and c = 1.
The complete function is y = –4x² + 8x + 1.
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