Answer:
A function is increasing when the gradient is positive
A function is decreasing when the gradient is negative
<u>Question 7</u>
If you draw a tangent to the curve in the interval x < -2 then the tangent will have a positive gradient, and so the function is increasing in this interval.
If you draw a tangent to the curve in the interval x > -2 then the tangent will have a negative gradient, and so the function is decreasing in this interval.
If you draw a tangent to the curve at the vertex of the parabola, it will be a horizontal line, and so the gradient at x = -2 will be zero.
The function is increasing when x < -2
![(- \infty,-2)](https://tex.z-dn.net/?f=%28-%20%5Cinfty%2C-2%29)
The function is decreasing when x > -2
![(-2, \infty)](https://tex.z-dn.net/?f=%28-2%2C%20%5Cinfty%29)
<u>Additional information</u>
We can actually determine the intervals where the function is increasing and decreasing by differentiating the function.
The equation of this graph is:
![f(x)=-2x^2-8x-8](https://tex.z-dn.net/?f=f%28x%29%3D-2x%5E2-8x-8)
![\implies f'(x)=-4x-8](https://tex.z-dn.net/?f=%5Cimplies%20f%27%28x%29%3D-4x-8)
The function is increasing when ![f'(x) > 0](https://tex.z-dn.net/?f=f%27%28x%29%20%3E%200)
![\implies -4x-8 > 0](https://tex.z-dn.net/?f=%5Cimplies%20-4x-8%20%3E%200)
![\implies -4x > 8](https://tex.z-dn.net/?f=%5Cimplies%20-4x%20%3E%208)
![\implies x < -2](https://tex.z-dn.net/?f=%5Cimplies%20x%20%3C%20-2)
The function is decreasing when ![f'(x) < 0](https://tex.z-dn.net/?f=f%27%28x%29%20%3C%200)
![\implies -4x-8 < 0](https://tex.z-dn.net/?f=%5Cimplies%20-4x-8%20%3C%200)
![\implies -4x < 8](https://tex.z-dn.net/?f=%5Cimplies%20-4x%20%3C%208)
![\implies x > -2](https://tex.z-dn.net/?f=%5Cimplies%20x%20%3E%20-2)
This concurs with the observations made from the graph.
<u>Question 8</u>
This is a straight line graph. The gradient is negative, so:
The function is decreasing for all real values of x
![(- \infty,+ \infty)](https://tex.z-dn.net/?f=%28-%20%5Cinfty%2C%2B%20%5Cinfty%29)
But if they want the interval for the grid only, it would be -4 ≤ x ≤ 1
![[-4,1]](https://tex.z-dn.net/?f=%5B-4%2C1%5D)
<u>Question 9</u>
If you draw a tangent to the curve in the interval x < -1 then the tangent will have a negative gradient, and so the function is decreasing in this interval.
If you draw a tangent to the curve in the interval x > -1 then the tangent will have a positive gradient, and so the function is increasing in this interval.
If you draw a tangent to the curve at the vertex of the parabola, it will be a horizontal line, and so the gradient at x = -1 will be zero.
The function is decreasing when x < -1
![(- \infty,-1)](https://tex.z-dn.net/?f=%28-%20%5Cinfty%2C-1%29)
The function is increasing when x > -1
![(-1, \infty)](https://tex.z-dn.net/?f=%28-1%2C%20%5Cinfty%29)