Question

Graphs of Function,

Answer 1

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

$(- \infty,-2)$

The function is decreasing when x > -2

$(-2, \infty)$

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:

$f(x)=-2x^2-8x-8$

$\implies f'(x)=-4x-8$

The function is increasing when $f'(x) > 0$

$\implies -4x-8 > 0$

$\implies -4x > 8$

$\implies x < -2$

The function is decreasing when $f'(x) < 0$

$\implies -4x-8 < 0$

$\implies -4x < 8$

$\implies x > -2$

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

$(- \infty,+ \infty)$

But if they want the interval for the grid only, it would be -4 ≤ x ≤ 1

$[-4,1]$

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

$(- \infty,-1)$

The function is increasing when x > -1

$(-1, \infty)$