Answer:
Below.
Step-by-step explanation:
a) First we find the derivative of the function, which gives us the gradient in terms of x:-
y = x^2 + 4x + 3
Gradient at x = dy/dx = 2x + 4.
When the curve cuts the x-axis, y = 0:
x^2 + 4x + 3 = 0
(x + 1)(x + 3) = 0
so x = -1 and -3 where the curve cuts the x axis.
The gradients at these point are therefore:
At x = -1 gradient = 2(-1) + 4 = 2
at x = -3 gradient = 2(-3) + 4 = -2.
b) (i) Where the tangent is parallel to x axis the gradient = 0
so 2x + 4 = 0 giving x = -2.
We now need to substitute x = -2 into the original function to find the y coordinate.
So the coordinates are (-2, (-2)^2 + 4(-2) +3))
= (-2, -1)
(ii) First find the slope of the line 6x + 3y = 7:
3y = -6x + 7
y = -2x + 7/3.
So the slope is -2.
Thus -2 = 2x + 4
2x = -6
x = -3
So the coordinates are (-3, (-3)^2 + 4(-3) + 3)
= (-3, 0).
