(a) The function is increasing over the intervals (-∞, -5), (-2, 2), and (6, ∞)
(b) The function has local minima at x = -2, 6.
(c) The sign of the function's leading coefficient is positive.
(d) A possibility for the degree of the function is 5.
Part a,
For any given function, y = f(x), if the output value is increasing when the input value is increased, then, the function is generally referred to as an increasing function.
By critically observing the graph of this polynomial function above, we can logically deduce that it is increasing over the following intervals;
increasing = (-∞, -5), (-2, 2), and (6, ∞).
Part b.
The local minimum refers to the x-value at which the derivative of a function gives the minimum output value. Hence, a local minima is a point that has the least y-value, compared to other surrounding points.
Therefore, this polynomial function has local minima at the following points;
(-2, -1) ⇒ x = -2
(6, -3) ⇒ x = 6.
Part c.
Since this polynomial function falls to the left and rise to the right, then it is an odd degree function with a positive leading coefficient.
Part d.
The degree of a polynomial function is the greatest exponent (leading coefficient) of each of its term. If an x-intercept passes through the x-axis, then the zeros has a multiplicity of 1.
Based on the graph, the x-intercepts are -6, -3.2, -0.8, 4, and 7.1, so a possibility for the degree of this function is 5.