Question

use the graph to answer the following questions. (a) over which intervals is the function increasing? choose all that apply. (-∞, -5) (-5, -2) (-2, 2) (5, 8) (2, 8) (8, ∞) (b) at which x - values does the function have local minima? if there is more than one value, separate them with commas. -5, 2 (c) what is the sign of the functions leading coefficient? positive (d) which of the following is a possibility for the degree of the function? choose all that apply. 4 5 6 7 8 9

Explanation:

Step1: Determine increasing intervals

A function is increasing when the graph goes up from left - to - right. By observing the graph, we see it is increasing on (-5, -2) and (2, 8).

Step2: Identify local minima

Local minima occur at points where the function changes from decreasing to increasing. From the graph, we can see local minima at x = - 5 and x = 2.

Step3: Find sign of leading coefficient

Since the function goes up on the far - right side (as x approaches positive infinity), the leading coefficient is positive.

Step4: Determine possible degrees

The number of local extrema (local maxima and minima) of a polynomial function is at most one less than the degree of the polynomial. Here we have 2 local minima and potentially some local maxima. A polynomial of odd degree has opposite end - behaviors. Since our function has opposite end - behaviors (down on the left and up on the right) and 2 local minima, possible odd degrees greater than 2 are 5, 7, 9.

Answer:

(a) (-5, -2), (2, 8)
(b) -5, 2
(c) Positive
(d) 5, 7, 9