Question

Show all steps to determine the vertex for y= x2 - 6x+2

Answer 1

Vertex of the given equation is (3,-7).

Explanation:

We have given a quadratic equation in standard form.

y = x²-6x+2

We have to find the vertex of the given equation.

y = (x-h)²+k is the vertex form of the given equation where (h,k) is the vertex of the equation.

Adding and subtracting (-3)² from given equation.

y = x²-6x+(-3)²+2-(-3)²

Making formula: (a-b)² = a²-2ab+b²

y = (x-3)²+2-9

y = (x-3)²-7

Comparing above equation with vertex form , we have

h = 3 and k = -7

Hence, vertex of the given equation is (3,-7).

Answer 2

The vertex of the given conic section is (3, -7)

Explanation:

The function given represents a parabola which is part of the family of conic sections. The vertex of a parabola represents the turning point of the curve, that is the slope of the line passing through the vertex is 0.

We first determine the gradient function of the curve by applying differential calculus. We differentiate the function with respect to x then set the result to 0;

dy/dx = 2x - 6; the gradient function.

Setting to 0 and solving for x;

2x - 6=0

2x = 6

x = 3

Lastly, we substitute x =3 in the equation to determine the y-value;

y = 3^2 -6(3)+2

y = -7

The vertex of the parabola is thus;

(3, -7)

See the attachment below