#y=4x^2-5x-1# is a quadratic formula in standard form:
#ax^2+bx+c#,
where:
#a=4#, #b=-5#, and #c=-1#
The vertex form of a quadratic equation is:
#y=a(x-h)^2+k#,
where:
#h# is the axis of symmetry and #(h,k)# is the vertex.
The line #x=h# is the axis of symmetry. Calculate #(h)# according to the following formula, using values from the standard form:
#h=(-b)/(2a)#
#h=(-(-5))/(2*4)#
#h=5/8#
Substitute #k# for #y#, and insert the value of #h# for #x# in the standard form.
#k=4(5/8)^2-5(5/8)-1#
Simplify.
#k=4(25/64)-25/8-1#
Simplify.
#k=100/64-25/8-1#
Multiply #-25/8# and #-1# by an equivalent fraction that will make their denominators #64#.
#k=100/64-25/8(8/8)-1xx64/64#
#k=100/64-200/64-64/64#
Combine the numerators over the denominator.
#k=(100-200-64)/64#
#k=-164/64#
Reduce the fraction by dividing the numerator and denominator by #4#.
#k=(-164-:4)/(64-:)#
#k=-41/16#
Summary
#h=5/8#
#k=-41/16#
Vertex Form
#y=4(x-5/8)^2-41/16#
graph{y=4x^2-5x-1 [-10, 10, -5, 5]}
