Question

151b

1. find the vertex of each quadratic function.

(1) $y=x^{2}-8x-3$
(3) $y=x^{2}+2x-3$
sol
$y=(x^{2}-8x)-3$
$=(x^{2}-8x+\square)-\square-3$
(2) $y=x^{2}-4x+6$
(4) $y=x^{2}+6x$

Explanation:

Step1: Complete the square for (1)

$y=(x^2-8x+16)-16-3=(x-4)^2-19$

Step2: Identify vertex for (1)

Vertex form $y=a(x-h)^2+k$ gives $(h,k)=(4, -19)$

Step3: Complete the square for (2)

$y=(x^2-4x+4)-4+6=(x-2)^2+2$

Step4: Identify vertex for (2)

Vertex form gives $(h,k)=(2, 2)$

Step5: Complete the square for (3)

$y=(x^2+2x+1)-1-3=(x+1)^2-4$

Step6: Identify vertex for (3)

Vertex form gives $(h,k)=(-1, -4)$

Step7: Complete the square for (4)

$y=(x^2+6x+9)-9=(x+3)^2-9$

Step8: Identify vertex for (4)

Vertex form gives $(h,k)=(-3, -9)$

Answer:

(1) Vertex: $(4, -19)$
(2) Vertex: $(2, 2)$
(3) Vertex: $(-1, -4)$
(4) Vertex: $(-3, -9)$