Finding Equation Of A Curve From Its Gradient Function

Example 1:
Find the equation of the curve that has the gradient function  $\frac{dy}{dx}=2x+8$ and passes through the point (2, 3).

Solution:

y = x² + 8x + C

3 = 2² +8(2) + C  (2, 3)

C = –17
Hence, the equation of the curve is
 y = x² + 8x – 17

The gradient function of a curve is given by 2x – 4 and the curve has a minimum value of 3. Find the equation of the curve.

At the point where a curve has a minimum value,  $\frac{dy}{dx}=0$
$\frac{dy}{dx}=0$ 

2x – 4 = 0

x = 2

When x = 2, y = 3.

3 = 2² – 4(2) + c

c = 7

Hence, the equation of the curve is