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.