The important points for sketching a quadratic are where it crosses or turns at the #x# axis, crosses the #y# axis and the vertex of the parabola.
It crosses the #x# axis at the roots:
#3x^2 + 5x - 2 = 0#
Factoring:
#( 3x -1 )( x + 2) = 0 => x = 1/3 and x = -2#
It crosses the #y# axis where #x = 0#:
#3(0)^2 + 5(0) -2 = 0 => y = -2#
Vertex is at# ( -5/6 , -47/36 )#
To find vertex of #3x^2 + 5x -2 #:
Place a bracket around the terms containing the variable.
(#3x^2 + 5x)- 2# factor out the coefficient of #x^2# if this is not#
1#
#3(x^2 + 5/3x) - 2#
Add the square of half the coefficient of #x# inside the bracket and subtract it outside the bracket.
#3(x^2 + 5/3x + (5/6)^2 ) - (5/6)^2 - 2#
Arrange
#(x^2 + 5/3x + (5/6)^2 )# into the square of a binomial and simplify# - (5/6)^2 - 2#:
#3( x + 5/6 )^2 - 47/36#
This is now in the form #a( x - h )^2 + k#
Where #h# is the axis of symmetry and #k# is the maximum or minimum value of the function.
So vertex is #( -5/6 , -47/36 )#
Points for drawing graph are:
#( 1/3 , 0 ) ( -2 , 0 )# roots of equation.
#( 0 , -2 )# #y# axis intercept.
#(- 5/6 , - 47/36 )# vertex
