Question

use the graph to answer the following questions. (a) over which intervals is the function increasing? choose all that apply. (-∞, -8) (-8, -6) (-2, 3) (-6, 3) (7, 9) (9, ∞) (b) at which x - values does the function have local maxima? if there is more than one value, separate them with commas. (c) what is the sign of the functions leading coefficient? select (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: Identify increasing intervals

A function is increasing when the graph goes up from left - to - right. By observing the graph, the function is increasing on the intervals $(-8,-6)$, $(-2,3)$ and $(7,9)$.

Step2: Determine local maxima

Local maxima occur at the peaks of the graph. From the graph, the $x$ - values of local maxima are $x=-6,x = 3,x=9$.

Step3: Analyze end - behavior for leading coefficient

As $x\to\pm\infty$, the function goes to $-\infty$. For a polynomial function $y = a_nx^n+\cdots+a_0$, if $n$ is even and $a_n<0$ or $n$ is odd and $a_n < 0$, the function has this end - behavior. So the sign of the leading coefficient is negative.

Step4: Estimate degree from turning points

The number of turning points of a polynomial function $y=f(x)$ is at most $n - 1$, where $n$ is the degree of the polynomial. The graph has 5 turning points, so the degree $n$ satisfies $n-1\geq5$, or $n\geq6$. Also, since the end - behavior is that as $x\to\pm\infty,y\to-\infty$, the degree must be even. So possible degrees are 6 and 8.

Answer:

(a) $(-8,-6),(-2,3),(7,9)$
(b) $-6,3,9$
(c) Negative
(d) 6, 8