By gradient, we mean the slope. So we want the point where the slope is 8. The derivative of a function is its slope, so we have to find the derivative for y.

y = (x - 3)(x + 5)

y' = (x - 3)'(x + 5) + (x - 3)(x + 5)'

y' = (1)(x + 5) + (x - 3)(1)

y' = (x + 5) + (x - 3)

y' = 2x + 2

So, since the derivative is the slope, we can find where this equation equals 8.

8 = 2x + 2

6 = 2x

3 = x

Now put that into our original equation to find y:

y = (x - 3)(x + 5)

y = (3 - 3)(3 + 5)

y = (0)(8)

y = 0

That leaves our answer (3,0).

f(x) = (x - 3)(x + 5)

You can get the gradient if you calculate the derivate of the function.

The function looks like: u.v → the derivate will be like: (u'v + v'u) → where:

u = (x - 3) → u' = 1

v = (x + 5) → v' = 1

f'(x) = (u'v + v'u)

f'(x) = (x + 5) + (x - 3)

f'(x) = x + 5 + x - 3

f'(x) = 2x + 2

To find when the gradient is 8, we have to solve the equation: f'(x) = 8

2x + 2 = 8

2x = 6

x = 3 → that is the abscissa of the point

f(x) = (x - 3)(x + 5)

f(3) = (3 - 3)(3 + 5) = 0

The coordinates of the point where the gradient is 8 are (3 ; 0)