what is the slope of the curve x^2 + 3 = 4y

Answer 1

x^2 + 3 = 4y

solve for y

y = x^2/4 +3/4

take the derivative

y' = x/2 = slope at a point (x,y)

= (x, f(x)) = (sqr(4y-3),y) or (x, x^2/4+3/4)

the slope of the parabola is the slope of a tangent line at a point on the parabola

x^2 +3 +4y is an upward opening parabola with vertex=minimum point = (0, 3/4)

for all x coordinates <0, the slope is negative,

for all x coordinates >0 the slope is positive

at the vertex with x=0, the slope = 0