Answer:
A function is increasing when the gradient is positive
A function is decreasing when the gradient is negative
Question 7
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
[tex](- \infty,-2)[/tex]
The function is decreasing when x > -2
[tex](-2, \infty)[/tex]
Additional information
We can actually determine the intervals where the function is increasing and decreasing by differentiating the function.
The equation of this graph is:
[tex]f(x)=-2x^2-8x-8[/tex]
[tex]\implies f'(x)=-4x-8[/tex]
The function is increasing when [tex]f'(x) > 0[/tex]
[tex]\implies -4x-8 > 0[/tex]
[tex]\implies -4x > 8[/tex]
[tex]\implies x < -2[/tex]
The function is decreasing when [tex]f'(x) < 0[/tex]
[tex]\implies -4x-8 < 0[/tex]
[tex]\implies -4x < 8[/tex]
[tex]\implies x > -2[/tex]
This concurs with the observations made from the graph.
Question 8
This is a straight line graph. The gradient is negative, so:
The function is decreasing for all real values of x
[tex](- \infty,+ \infty)[/tex]
But if they want the interval for the grid only, it would be -4 ≤ x ≤ 1
[tex][-4,1][/tex]
Question 9
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
[tex](- \infty,-1)[/tex]
The function is increasing when x > -1
[tex](-1, \infty)[/tex]
