Question

Find the coordinates of vertex for the parabola given by the equation f(x)=3x^2-x-1

Answer 1

The vertex of a parabola:

y = a x ^ 2 + b x + c

is the point where the parabola crosses its axis.

If the coefficient of the x ^ 2 term is positive, the vertex will be the lowest point on the graph, the point at the bottom of the “U”-shape.

If the coefficient of the x ^ 2 term is negative, the vertex will be the highest point on the graph, the point at the top of the “U”-shape.

The expression : - b / 2 a

gives the x-coordinate of the vertex.

In this case :

a = 3 , b = - 1 , c = - 1

The x - coordinate of the vertex :

x = - b / 2 a = - ( - 1 ) / 2 * 3 = 1 / 6


The y-coordinate of the vertex :

y = 3 x ^ 2 - x - 1

y = 3 * ( 1 / 6 ) ^ 2 - 1 / 6 - 1

y = 3 * 1 / 36 - 1 / 6 - 1

y = 3 / 36 - 1 / 6 - 1

y = 3 / 3 * 12 - 1 / 6 - 1

y = 1 / 12 - 1 / 6 - 1

y = 1 / 12 - 2 / 12 - 12 / 12

y = - 13 / 12

The coordinates of the vertex :

( 1 / 6 , - 13 / 12 )