find the values of a and b if the graph of the function y ax2 bx 3 has | TutorVista

Question: Find the values of 'a' and 'b', if the graph of the function y = ax² + bx + 3 has a vertex at (2, - 1).

A ) a = 1; b = - 4
B ) a = 4; b = - 1
C ) a = - 4; b = 1
D ) a = - 1; b = 4

Steps to derive

1 y = ax² + bx + 3

2 a = a, b = b, c = 3
[Compare with y = ax² + bx + c.]

3 The vertex is (2, - 1).

4 x-coordinate of the vertex, h = - b2a = 2

5 ⇒ b = - 4a
[Multiply throughout by - 2a.]

6 y = ax² + (- 4a)x + 3
[Substitute the values.]

7 ⇒ y = ax² - 4ax + 3

8 The vertex of a parabola is a point on it. So, it will satisfy the equation of a parabola.

9 Substitute (2, - 1) in y = ax² - 4ax + 3 to find the value of â€˜aâ€™.

10 - 1 = a (2)² - 4a(2) + 3

11 ⇒ - 1 = 4a - 8a + 3

12 a = 1
[Simplify.]

13 Substitute a = 1 in b = - 4a, to find â€˜bâ€™.

14 b = - 4(1)

15 ⇒ b = - 4

16 So, the equation becomes y = x² - 4x + 3.

Hence the right answer is Option A

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